Lots of people are getting out of their cars and onto bikes these days, because of the high cost of gasoline , parking hassles, and concerns about staying physically fit. When natural disasters or terrorist acts strike, people repeatedly find that a good bicycle is a fine thing to have. On a "bang for the buck" basis, the bicycle is one of the finest travel machines Mankind has devised. When the motor vehicle was still in its infancy, armies the world over were putting their troops on bicycles. The armies knew they could move a lot of soldiers, with gear, impressive distances in impressively short times.
I highly recommend bicycles as part of your preps. A bike for each member of your family or group is best, and having ridden around on them quite a bit, I have some definite opinions on what's best to look for in a "survival bike". It should be durable, comfortable, versatile, and easy to work on. I should note that my experience is in the US, so if you're outside the US you might want to translate what I recommend to your local area. But I think the kind of bike I recommend is fairly universal.
First, any bike is better than no bike at all. Secondly, your survival bike should be one you're familiar with, ride often, and are comfortable on. It will be your every-day, or at least one of your everyday bikes. If you have more than one it may be your "winter bike", or the one you do grocery errands on, or putt around on your local trails or unpaved roads. Consider it a mule who's an old friend who you can always count on.
So, what's best to get? Let's work from the wheels up. These days you'll see a lot of skinny bikes with skinny wheels and little, skinny, tire valves. These skinny ones are called Presta, generally need a different pump head, are more complicated to use, and are generally on skinny wheels on the kind of bikes I don't favor. We don't need to win a bike race here, and we don't need European-standard valves. I also say avoid the old skinny-tire 10-speeds, the one your Dad may have bought in the 1960s and left you, for instance. These have Schrader-valve wheels, but they're an older standard for "vintage" skinny-tire bikes, and tires and tubes can be hard to find, in limited choices, these days. Leave all this skinny-tire stuff to the racers, messenger kids, and vintage-bike enthusiasts. Stick with the Schrader valve, the Schrader valve is found on car tires, pumps that fit it are found everywhere, and in the US it's by far the most common, proven, and user-friendly valve type.
What I recommend is, you get a bike that has 26" wheels. That's the standard in the US and is found on cruisers, a lot of "city" or "commuter" bikes, and on the tons and tons of mountain bikes that are out there. You want something you can get tires and tubes for everywhere, fill up just about anywhere, and there are pumps widely available.
An older mountain bike is what I recommend. What goes under the name "mountain bike" these days is most often something I'd avoid. The reason is, almost all of them have suspension, springs and shock absorbers, on them. Those are to be avoided. They have their place, but on a general-purpose bike all they do is make the ride mushy, wasting your pedal-power, make the bike heavier, and add complication and expense. Sure, they cushion the bumps, but that's what pneumatic tires were invented for. The high-end bikes are very expensive, and made for "downhill" riding, going fast and bouncing over stuff. You won't do that with your bike, you'll lift it over that log etc. The big-box store cheapo mountain bikes are made to look "hi-tech" and are heavy, inefficient, and really not much fun to ride. Any big-box store bike isn't going to last, and at their low cost they're still too expensive. This is why I like older, "hard-tail", mountain bikes. When mountain bikes were a new thing, people were willing to pay a premium price for them. Also, the manufacturers weren't sure how roughly they'd be treated, so they tended to build them really well. This was the age of quality steel frames (look for Cromoly or Cro-Mo, on a sticker on the frame) with lugged construction which means at every joining point, the steel is double-thickness. Look up "lugged bicycle frame" on a search engine's images, it's a very handy thing to know how to identify.
These older, non-suspension mountain bikes have often been living in garages for a decade or two, and since a lot of people don't appreciate what they are, you can get them reasonably. They often have stainless-steel spokes on the wheels, and often the wheels are made by Araya, a Japanese wheel maker well-known for making motorcycle wheels. These are signs of quality to look for. In fact as a rule-of-thumb, if the spokes aren't stainless-steel, pass on it. Stainless spokes will have a dullish shine and feel smooth when you run your fingers down them.
I recommend garage sales and thrift stores and so on for price, over Internet sites because I feel the prices trend high on sites where someone has to go through some effort to list it and describe it, and likewise there's a large, well-informed public scanning the ads. You want the bike that gets pulled out for the garage sale "because we've had that old thing forever" and so on. Just remember: look for quality.
You'll want to look into bike fit, seats, various rack and pannier systems, etc. You can go fancy on racks, but the humble folding wire jobs that hang off of a rear rack are better than nothing by far. At the high end you have Ortlieb panniers, and then there are many types of homemade panniers made out of buckets and ammo boxes and so on. There are lots of plans online. A basket on the front looks a bit nerdy but they're extremely useful. Put one on and you'll wonder how you got by without it. Trailers, and those made by Burley are generally the best, are an additional thing. You can carry 100 pounds in a Burley Nomad, for instance. Again, look for them used, as they're quite pricey new.
Now some more about tires and tubes and wheels. After all, they really are where the rubber meets the road! First, you may have heard of a product called Slime. If you have "goat head" (Tribulus terrestris) weeds growing in your area, Slime is going to be a must-have. You can get tubes with it already installed, or you can put it in - the directions on the bottle are very easy to follow. You should know how to fix a flat anyway, and 26" wheels seem to be about the easiest to work on, as far as changing out tires and tubes. I recommend learning how to patch a tube, using the old-school patch kit with the "vulcanizing" cement in the little tube. Tubes are expensive these days! I'm about to go back to my old rule from my college days: Re-tire a tube after three patches.
Used bicycle tubes are extremely useful for all kinds of uses, so don't throw 'em out. For tires, with Slime, the tires you get with the bike should be fine, assuming they're not old and dried out (look for cracking in the sides of the tires). There are some highly regarded tires with Kevlar in them for hard usage like touring, or police-bicycle work, like the Schwalbe Marathon. If you've got to have "the" tires and have the budget, by all means get 'em. But you can get tires in all price ranges. Don't forget rim-bands, which are little strips of rubber or plastic that cover the nipples (bases) of the spokes inside the rim - uncovered, those will eventually wear through the tube and give you a flat. Remember that the tube needs a little TLC; if you get a flat, you must remember to check the tire to make sure the thorn, piece of glass, etc., that caused the flat isn't still in the tire.
You can buy bike stuff in a bike shop of course, and it's good to patronize your local bike shop just like any small business. But if you're on a budget or stocking up, that big 'Mart can't be beat. Or your local hardware store. There's a large population of people who go around by bike and are on a budget, and "dime stores" and their descendants generally have a bike department with basic tires, tubes, lights, all the things utilitarian riders need.
I suggest stocking up things that wear, like tires, tubes, grips, rim-bands, seats, pedals, cables, brake pads, all kinds of "consumables". Tires don't store well in the sun, so a dark part of the garage is much better. The rubber actually ages just from exposure to air, so if you are really serious you can put a bunch of tires, tubes, and rim bands in a 55-gallon drum or something with nitrogen (sold at car-parts places to fill tires with) or at least an oxygen absorber.
I want you to get the best bang for your buck, so I really suggest you check garage sales, church sales, places like that for bikes and parts. Lots of small things like a decent seat ... that'll run you a minimum of $20 at a bike shop and often quite a bit more, are often found looking for a home at a garage or church sale for a few dollars. Grips, tubes, really every little part, will show up at bargain-basement prices. What I'm leery of and think you should be, is the large commercially-run "swap meet" or "flea market" because a lot of stolen bikes show up at those. You can being a smartphone and check against the listings on the National Bike Registry (NationalBikeRegistry.com) but what if the owner didn't register theirs? You just can't tell. One suggestion is to get a bill of sale and take a photo of the seller's driver's license, and if they won't let you do that, steer clear. Be careful in the jungle of deals-too-good-to-be-true.
Helmets are a personal choice in most areas, also in a lot of areas they're not a choice if you're a minor. I'm not going to recommend buying a used "lid", fortunately there are a lot of them out there new at reasonable prices. A more expensive helmet may be lighter, cooler in hot weather, or the one worn by this year's World champion, but it's not necessarily any safer than a sensibly-priced one. The one opinion I have about helmets is, if you wear one, might as well get a light-colored one, like yellow. It will increase your visibility to drivers, although in tough times you may not want to be seen so easily. That's when you get out the camo tape.
Lastly, if you have to visit one bike site online, check out Sheldon Brown's site. Sheldon has passed and will be missed, but his bike shop has kept his work online because it's so helpful, friendly, and comprehensive. In fact it can be almost overwhelming so the other resources I recommend are videos on YouTube, and classes, generally free, held by your local bike shop or bike club.
Short and sweet:
* Bikes have moved armies, officially and un-officially. They can move you.
* 26" wheels, with Schrader valves, by far the most common in the US; that's desirable.
* Get an older mountain bike, no suspension, Cro-Moly frame, stainless steel spokes.
* Learn to use Slime
* You can build a fine stable of bikes from thrift, garage, and church sales.
* Stock up on consumables, like tubes and tires.
* There are a few things you always buy new. Bike helmets are one of them.
* Learn more from Sheldon Brown, YouTube, local shops, clubs, and groups.