ds had a 12" one before he was 2 (he surprised us all by just climbing on one belong to one of our neighbours and riding it away when he was just 20 months!). Now at rising 4 we've just bought him a 16" one as he had the seatpost of the old one about 2" abouve the maximum line. 16" is probably a tiny bit big for him at the moment, but even the 14" would have been too small by about Christmas this year.
For a 3 year old I'd get 12" if they're small/average, or if she's tall for 3 I'd try out some 14" ones in your local shops for size, as they might last her a little longer.
I second Charliecat's suggestion of Sterling House as a good place to buy kids bikes at a reasonable price.
Dd had a bike for her 3rd birthday, which was an April birthday, so just in time for the summer. I've been wondering what to do about ds, who'll be 3 in January - not a great time for bikes really, UNLESS, you do what we're about to do, and head out to Australia, which means a bike will be perfect. My winter baby gets a summer birthday!
We got dd's bike in Halfords. Boring, and expensive, judging by Sterling House
Argos have the 12" Barbie bike going for 1/2 price at the moment. My dd will also be 3 in september. We've bought that for her. She's been asking for one for ages. Ever since she saw the same one in Halfords at twice the price!
DD, now 2, surprised us by riding a friend's bike around 4 months ago. We bought her first big bike recently on Ebay for £5.00. It's in perfect condition and ideal for a first bike as it has already had a few scrapes. We saw the same one in Halfords for £59.99. It's a 12" but I can see she will be on a bigger one in a year's time or so as she's quite tall. Another thought is that some primary schools hold bike sales for parents to sell bikes which have been outgrown - worth enquiring if you can wait a while.
We bought DD This Rayliegh Mini Miss bicycle for her 3rd birthday. It cost us just under £80 from an independent bike shop locally. It has 14 inch wheels, and trainer wheels attached. She really likes it.
i think what you also have to think about,not getting a bike too big that she can not learn to ride without stabilizers.my dd got a 12" one for her 3rd birthday she learnt to ride this without stabilizers at 5yrs.Alot of her friends got bigger bikes but cannot ride them without stabilizers.We bought her new one in toys r us,we got a 16" the shop assistant tried to sell me a 18".I knew if we got it too big she would not ride it.
Anyone bought a Raleigh Lily 14" (Argos stock it @£49.99) ? We've seen the Raleigh Molly ones can't face the furry saddle cover, tassles etc especially since the Lily looks basically the same minus the gimmicks but these are £20-30 more ! Mini Miss seems a bit heavy for the size although dd quite liked it.