I'm still a bit of a rookie but here is my advice for what it's worth:
First, decide what kind of motorized bicycle you want. Your choices are gas or electric, chain drive or friction drive, 2 stroke or 4 stroke, etc. Determine how much you have to spend, "cheap" can mean many different things to many different people. If you want the "cheapest
" than go for a Chinese made 2 stroke, center mount, chain drive kit. They are the most common for beginners. I think the cheapest ones can be found at ebay.com.
Next, go on craigslist.org or Walmart or something to find a cheap bike. Beach cruisers work good but consider that you will need brakes and I wouldn't rely on using just coaster brakes only. I recommend getting brakes for both wheels, standard V-brakes or disc brakes. Make certain that the bike has enough room in the center to mount the engine. If you get one with really thick tubes you will need to make special accommodations to mount the engine. You can make your own or purchase a universal mount. Search this site, there are many threads covering bicycle selection.
Next, get a 2 stroke engine kit. There are several different sizes of engines available, 66/80cc is very common but there are many to choose from. Some vendors are good and others, not so much. Make certain that the kit you choose has all the following standard parts: engine, carburetor, ignition coil & CDI, throttle, throttle cable, clutch cable, clutch lever, chain, chain cover, 44T sprocket W/ 9 hole clamp pads, chain tensioner, fuel tank, petcock valve, and exhaust pipe. These cheap small engines only last a certain number of miles before failure so make certain it is a new engine
and not used one. Most of these cheap kits are made in China so you should expect to get poorly made parts such as screws and nuts. They don't make very good steel in China so it is smart to replace all the screws and nuts with American made ones from the beginning. Also, some parts are made perfect to the specs and others are way off. No two engines come off the line the exact same. I think that quality control in China is just a myth. You should expect to purchase a high quality spark plug and not use the one in the kit. Make certain that you inspect the engine and all parts for defects before assembly.
Next, assemble the kit on to the bike. You don't have to be a mechanic but you MUST be able to turn a screwdriver and a wrench. You must be somewhat mechanically inclined. Go to youtube.com and search "motorized bicycle kit" for video assembly instructions. When you have questions, and you will, come here and use the search function to find the answers. Most of your questions have already been asked and answered here. There are some pros here who have built many motorized bicycles and seem happy to help answer questions but it is best to research things yourself first. Expect to spend time on maintenance and to travel with tools for roadside fixes. Motorized bicycles create a lot of vibration which makes them require constant tightening of screws and nuts, much more than a moped. Make certain that you mix the oil in the gas at the correct ratio. There are many upgrades you can get. Make certain that you research each of them before you buy. I would say that the best upgrade I have gotten so far is an expansion pipe from sickbikeparts.com, it was well worth getting.
Motorized bicycles are super fun and can be great forms of transportation but if you hate to have to tinker with it alot then I urge you NOT
to get a motorized bicycle, remember, it's a hobby not a Harley.
Hope that helps, good luck!