Setting up a mobile phone number
Like the US, the UK offers various different phone providers where you can register for both pre-pay sim cards or post-pay contracts.
Three (3), EE, O2, Vodafone
PRE-PAY ("pay as you go")
This is where you would get a sim-card and pay for your usage in advance.
This is what I currently use and I spend about 15GBP/month for 300 minutes and 5GB of Internet. If we go to place that has WIFI, most places do, then I always use the local WiFi to reserve my data.
I use Three and they have an app that you can use to top-up and then turn the "top-up" into an "add-on"(the shop can help you do this the first time).
Vodafone also has lots of great pre-pay options.
Check out each of the companies and see which one fits your needs best.
Here is an example from Three on how to set-up a new SIM
2. PAY MONTHLY (contract)
Here you will be offered a package that includes a phone. You will have to pay an amount up front plus a set monthly fee for however long your contract is for (usually 2 years). The amount you pay-per-month will vary on provider as well as and the phone and the package's benefits. Usually, after two years, you will be eligible to get a new phone/change your package.
What to bring with you:
If you're short on time, you don't have to go into a shop to purchase a package! Most providers give you the option to set yourself up with a phone and contract through their website.
Do note that you will be subject to a credit check. Also, it is safe to have ready your proof of address (bill or council tax) and your ID (passport, drivers license).
If you are like me and have a phone already, you may want to do pre-pay at first.
Students often get discounted rates, so if you're a student, ask what discounts or special packages they offer