Hi Alison
We have always gone to Plakias early on in the season - usually towards the end of April and the early part of May - so I can vouch for the weather being good at this time of year although as others have said, it can get a bit cool in the evenings. We've only had rain, I think, on a couple of occasions and only once was it of the heavy variety.
As for tavernas to eat with small children, I would recommend Finikas which is a bit of a way out of town perhaps (although if you are staying in town at least the walk back is downhill). If coming from the direction of Plakias, Finikas is at the brow of the hill off down a track to the right, just at the point of the "Welcome to Plakias" sign. There are rooms attached to the taverna which are often block booked by Austrian companies. Usually there are a lot of families with young children staying there, so there tends to be a big group of kids playing together. The food is excellent at Finikas and the owners, Kostas & Maria, have a four year old and I think a younger child as well so they will be aware of the pitfalls of eating out with a little one.
Paligremnos on the beach is also good - they have a play area just to the side of the taverna although depending where you are sitting you may not have a good view of the kids playing there so that might be a problem if your child is very young.
Hope this helps
Karen