I am in Zone 9b and want to attract butterflies and dragonflies in my backyard.
What can I do and what plants do I need to grow to encourage them to stay around?
I've always thought of dragonflies as being water-loving - that is where you see them, near rivers and ponds. And their larvae are aquatic.
So you need a pond if you want dragonflies that aren't simply fly-throughs. Our current house has a pond and we get a number of varieties of dragonfly in the summer.
Yes we have mosquitoes, but as far as I can tell they are not from the pond itself, but we're in Texas and they're from the other usual sources (standing water, water trays, etc). We do have quite a few fish in the pond and these appear to keep on top of the mosquito larvae.
Do the fish eat dragonfly larvae? Possibly. Dragonfly larvae are meant to be quite ferocious creatures so they might be able to fight back. I have seen dragonflies lay eggs in the pond, but have yet to see any emerge (perhaps I haven't been looking close enough in the right places).
Attracting and keeping butterflies around depends on two things - the nectar rich plants they like, and the food plants for their larvae. If you just have nectar rich plants they like, then they will visit, but they won't be able to lay eggs unless there are also host plants.Some of those host plants may not be things you want in your garden - nettles are the host plant for Small Tortoiseshell butterfly eggs in the UK, but we don't all have room for a large nettle patch. Milkweed and asters are common host plants for butterfly eggs in the USA, so you might not mind those too much.
That said, here's some info on plants butterflies like - whether all are suitable for your zone you'd need to check http://www.almanac.com/content/plants-attract-butterflies
As for dragonflies, I've never seen those anywhere but over bodies of water, in particular, moving water like streams and rivers. Butterflies, though, also appreciate access to water, but it doesn't mean they won't visit if there is none.