A proper removal company will be very expensive.
The next option is the "man and van", which gives the advantage of someone else to help with the heavy lifting, but a) you'll still have to do a lot yourself and b) you have to beware of the tin-pot operations that are just some geezer with a transit as they probably won't be properly insured for any goods they carry.
This one is a big operation and has set prices per hour for whether you load or get them to help and a mileage rate.
http://www.van-plus.com/ This one, local to you I think, isn't quite so clear, but they look ok. At least they have a land line phone number on their website
http://www.manvancardiff.co.uk/But there have been some reports of dodgy man and van crews and have been known to drive off with peoples stuff and hold it for ransom, charging a "storage fee" when it wasn't needed, etc.
Would you let someone in a plain unmarked van drive off with your stuff?
Hiring a van yourself is easy. I've driven hired transits, T5s and a boxed Luton and it's fine. You just have to remember it's bigger than your car.
So long as you're good at packing, and stack things logically and get a few sensible mates to help, you'll be fine.
Also consider whether you have any stairs to negociate? Is it in a block of flats, or are the stairs going to be tricky to get a bed up? This is where an experienced helper is good.
