we've just bought a 'Co-pilot Limo' bike seat from http://www.edinburgh-bicycle.co.uk/ on a friend's recommendation. We used to be keen mbikers but haven't been out since before ds' arrival 10 months ago. Haven't tried it out yet though - it's been raining!
A good piece of advice I was given - After you have the seat fitted on your bike, go out for a trial run with something on there that weighs about the same as your child. I took bag with a few books in it. It gave me more confidence to do it "for real" once I had a bit of practice.
I recommend the Rhode Gear one - I used it from when my son was 9 months and it reclines a bit so you don't have to worry about them lolling about. The back and sides are also very tall giving them extra protection, and something to lean their head on - some of the bike seats around don't have them much to lean on if they fall asleep. My friend used hers until her son was four.
It unclips from the bike really easily if you are leaving your bike parked somewhere where it might get nicked like a tube station. So I used to leave mine at my son's nursery while I cycled on to the tube for work and parked my bike there. He really liked it.
I was really nervous about the thought that it might make the bike unstable - but it doesn't. It is not like cycling with a rucksack on - because the weight is held over the back wheel so it is fine. The only thing I would say is - don't leave your child in it when the bike is leaning against a wall unless you are within holding distance. I reckon the child moving about could make the bike fall over which wouldn't be very nice. This didn't happen to me but it could have done!
From Edinburgh bicycle co-op you can also get a sort of poncho that goes over the child and their seat to keep them dry if it rains. Get the more expensive one of the two they offer as apparently the cheaper one is only polythene and rips easily. They also sell adult cycling capes if you are prepared to forget all notions of looking attractive when you cycle in the rain!
The Rhode Gear seat is relatively expensive but you might pick one up second hand eg e-bay. The rack that clips it to the back is expensive too so make sure you get that if you buy a second hand seat. That rack will fit to any bike I would say.
The other option is having the child sitting in front of you - better for talking to them, but those seats are very small and aimed at an older child I would say because if they fell asleep they wouldn't have anything to lean against.
We've been happy with the cheap ones that Argos do (we do A LOT) of cycling. They're only about £30. You can sometimes pick up a 2nd hand seat on Ebay for about £20, including postage. Let's see, one of ours is a "Bikey" and the other is a "Polosport". HTH.
I would get a trailer if you do a lot of cycling, esp. in cold weather, they are much more comfy for the child(ren).
Definitely second the suggestion to put a big sack of spuds in the seat for a trial outing, to get you used to the weight. I did cycle-touring & camping so I didn't find it much to adapt to.