Ok, first, "bike" means a bicycle or a motorcycle. That being said, I will, however, always use "bike" as "bicycle". If you're looking for a bicycle in a bike shop, always ask for a bicycle, not a bike, otherwise you would get them confused. Moreover, I will only consider city bikes, they are far more comfortable and practical than mountain bikes, when riding in Cambridge only. Cambridge is flat, you don't even need more than one speed on a bike. And mud guards on mountain bikes are rare, fragile and less efficient.
It seems that most stores will refuse to sell you a cheap bike, because "cheap" sounds like "bad" for them. Maybe for all Cambridgians "cheap" sounds like "bad", I don't know, but when I said cheap, all I meant was "not ridiculously expensive for someone who only wants to ride it for 30 minutes a day in Cambridge, a flat city".
Most shops have brand new bicycles for 300 pounds or more. (Not less.)
Some shops have really cheap bikes (i.e., they actually look both new and cheap).
I visited Cycle King on Mill Road. They have nice bikes that are not too expensive (starting at about 150 pounds). And you can buy a bike and wait a few minutes for final service, then you leave on your new bike. *Not* all bike shops do that. For instance, for most bikes in the Station Cycles of Grand Arcade (SCGA), they have no stocks and you have to wait for at least 3 working days before you may collect your new bicycle.
I bought my first bike at Greg's Cycles on Mill Road. In my opinion, they're a bit expensive, but I also feel that I didn't pay too much for the bike that I got: it's just that I could have been happier with a cheaper bike.
There are plenty of other bike shops in Mill Road. If you plan on getting a bike there, you may visit Mill Road on Google Maps with Street View before actually getting there. And don't go there too late when all shops are closing.
There is a nice tiny shop with a few bicycles to sell. It's very weird because they also have sunbeds! They have both brand new bikes and second hand bikes. They offer cheap and efficient service. There, I bought a second bicycle with a few non-expensive accessories such as a front basket (and then a back basket). The address: 263A Newmarket Road. It's just near to the big Tesco superstore. (On Google Street View, it's "Michael's Nails" and the pictures were taken in 2008. Now it's "FantaSun".) For sure, if I have a problem with a bicycle, I'll go there to have it repaired.
The big advantage of second-hand bikes, apart from the lower price, *because yes, there is one*, is that it is less interesting to steal an old bike than a new one. Everyone I talked to said that there was a big chance of getting a bike stolen. Hence, if you're not planning on spending a *lot* of money in super-strong super-expensive locks, don't buy a new shiny bike but a second-hand bike instead.
I think that in France, a few years ago, it became illegal to sell bicycles without "final service", i.e., all that must be done in order for the bicycles to be ready to use. In the UK, it seems you may buy a bicycle without final service and some shops do charge for final service.
If you're about to become a member of the university of Cambridge, you'll soon have a card that will give you an immediate 10% discount in a lot of bicycle shops.