there is something called duckling dunk on a monday or tuesday(?) at fulwood leisure centre. it's in the little pool with floats and stuff in the water. its ok, but the the pool is rubbish, just a dull rectangle. priory pool in penwortham is colder but better.
There is also a small private pool on Durton lane in Fulwood called otters which runs swimming classes for lots of different age groups. I took my ds there for a while until i had learnt what to do with him myself. We now go to fulwood regularly on a sunday morning although this is not an organised class. Would recommend a course of 6 weeks at otters (should cost about £25) just to get you started and then just use the local pools.
Have you been to West View - also has a nice childs pool.
I have also been to Otters on Durton Lane and the instructor was really good. We only went for 6 weeks though as it was getting to winter and the pool and changing area was freezing! The changing facilities were also a bit of a non-event to be honest as you just had to go in one big room which got really crowded if there were a lot of you.
I wouldn't rule it out completely though as I'm sure it would be a lot nicer in the summer.
Update - Otters have recently finished totally altering the changing areas - theres two now, usually too hot rather than freezing, and the loos are much better.
When your kids are old enough for lessons without you in the water, its considerably more expensive than Fulwood but you get to sit in a nice cafe bit and can watch through the glass while you sip lattes.
For future reference, when your kids are about 8 and they and their friends are competent swimmers its a really good place for a birthday party.
The "swimbabes" session at Leyland leisure Centre is fantastic. I have been to a couple of different classes and this one is by far the best. I first took Imogen when she was 12 weeks old. The session is led by a couple of instructors. They help you introduce your baby to the water, get them kicking their legs, swimming with floats and dunking them!! Which was scary at first but now Imogen loves it and loves swimming under the water. It is everyday apart from Friday from 12-12.40ish and costs £2.80 per session. You have to get there early some days as it gets busy and they can only let so many in.