Fuel won't be much. My bike is a 125 (a moped is 50cc) and used about £7 to get to London from Nottingham, which is around 100 miles. Since a moped has a smaller engine it will use less fuel. It's impossible to say exactly because it does vary between bikes, and obviously with how far you travel, but fuel isn't the main expense you need to worry about.
Your CBT, which is the one day training course you need to do before you're allowed on the road, will cost around £100. You legally have to do a CBT so can't get out of this. If you already have a bike and gear it will be a bit cheaper. Some people don't pick up everything in a day so need to go back for a second try, and this will make it more expensive.
Go on Autotrader or somewhere and see how much mopeds cost. I have no idea. You don't want to buy a new one, as you'll be devastated when you throw it down the road. You can get them quite cheaply (I'd imagine £500 or so) but it may then be unreliable or need a lot of maintenance.
You need insurance. My bike insurance was £250ish with just my CBT (ie not a full bike licence), but I am a lot older and have had car experience. I'd guess yours would be about £600+, but that depends on the bike, where you live and loads of other factors.
You need bike kit. You can get a helmet for around £50, but it won't be as good as more expensive options. You also really need a jacket, boots, gloves and trousers. People do ride bikes in shorts and t shirts, but if you fall off you will get hurt.
This is my knee when I came off, and was wearing kevlar jeans with knee protection.