You may be a little bewildered by the range of prices and types of radio microphones, so here's a short introduction:
The quality and price ranges from 'karaoke' types costing £20 or less, right up to a thousand pounds. The main differences will be in range, reliability of the link (see below), robustness of the hardware, and the audio quality. With the cheapest, don't expect more than about 10 metres of range -- any more, and you'll get gaps in the coverage, which may at worst give you a brief 'shhhtt' noise rather than silence.
Go more expensive, and you'll get a 'diversity' receiver with your microphone. Diversity means two separate receivers in one box, with two separate antennas. The receiver that's getting the better signal is selected automatically -- so there's a good chance that if one receiver sees a gap in coverage, the other one is still working well. (But watch out -- some cheaper receiver boxes have two antennas but aren't diversity!)
There are four different types of transmitter. Hand-held transmitters have the microphone and transmitter in the same tube, which opens to allow you to fit batteries. Add-on transmitters fit on to a standard microphone, making it a radio mike, although the lump may not appear very elegant. Lapel microphones connect to a separate box that you can wear on your belt or in your pocket, and finally head-worn microphones use the same type of box. If you're using a lapel microphone, it's often worth getting a better microphone than the standard one that came with the kit, because it will pick up less clothing noise, and probably be tougher.
There are two legal radio bands in the UK that don't need a licence to use: the 'VHF' one between 173.7 and 175.1MHz (megahertz), and the 'UHF' one, 863 - 865MHz. The UHF one is a pan-European allocation, but note that other European countries also allocate other bands, so don't assume that a radio mic bought in Europe is legal in the UK -- check! Vendors should put the frequency of their radio mic on the eBay page! Naturally, if you are going to use more than one, each needs to have its own frequency. Some more sophisticated designs allow the user to select from a range of frequencies.
Radio mics all need a battery. Older designs mostly use a PP3 style 9V battery, and operate for a time between 3 and 10 hours from an alkaline, or rechargeable, battery. Newer types often use a pair of AA or even AAA cells and last for 5 to 10 hours. If you have to manage radio microphones a lot, it's worth getting rechargeable nickel metal hydride (NiMH) batteries, and an intelligent charger which indicates whether the battery is charging, or fully charged.