Well, there are a large number of shade trees that you could grow in your zone. I would probably recommend a eucalyptus tree. They grow incredibly fast(several feet per year) plus you get this unique scent from the foliage. Another good tree would be the southern magnolia, grows fast and makes large white scented flowers.

^ eucalyptus deglupta or rainbow eucalyptus would probably be the best if you decide to get a eucalyptus. Very colorful it looks like the bark on the tree was painted.

Neat idea with the eucalyptus! I'm going to look into that!

We have a big magnolia in the front yard, so we are looking for something different.

Thanks so much for the replies!

Fast and shade??? Sycamore! My parents had one in Tampa. It grew from sapling to maybe 50 feet in less than 20 years. Huge spreading tree!

Pros: super fast growing. Cool pale peel-y bark. Large attractive leaves. Lots of shade! Some yellow fall color (not spectacular in the South). (If you just want fall color, you could try red maple - fairly popular in central FL for color. Attractive but not as fast growing.)

Cons: spikey seed balls, neighbors complain about leaves falling in winter

if tree climbing is very important in your tree selection--you might want to consider that unless you buy and plant a large and expensive tree now; by the time an average size nursery tree (5-6' tall at planting) is large and sturdy enough to climb your kids may be out of the tree climbing stage.

I do not find sycamore seed balls particularly spiky. They can be large however but they crush pretty easily under foot.

Two summers ago I planted two plane trees (sycamore hybrids). One was the cultivar Columbia and the other was the more common Bloodgood. Columbia far outstripped Bloodgood in growth rate. I got a nice bit of shade out of a year and a half old tree! Both were planted from 15 gallon cans. It will be a few years before the limbs are thick enough for climbing.

Here in California, we had an early chilly spell and it made the sycamores color beautifully.

rapid growers for your area that I suggest looking in to are

tuliptree (not good for climbing)
sycamore (can be good for climbing)
shumard oak (no good for climbing)
sugarberry (no good for climbing)
American elm (no good for climbing)

One rapidly growing tree you can grow that is very good for climbing is ficus benjamina, however these trees are not very good for your yard and have a number of liabilities.

Metasequoia is great for climbing, and it grows very fast, too. Good shade tree, it should do well where you are.

Dawn redwood good for climbing?
I think they OP means for kids to climb. Not for professional climbers.

Weeping willow might be another thought. While they will grow quickly and kids should be able to climb them within 3-4 years, they don't seem to get so big as to endanger houses. Leaf fall color is yellow.
Incidentally we just had a 100' eucalyptus tree fall on someone's house here in Texas. Don't know the particulars.

Aggressive roots on the weeping willow though. And they drop quite a bit of litter.

I second the sweetgum. The florida native sweetgums develop much smaller spikey seed balls than the rest of the country. There's also a fruitless cultivar called rounditaloba. Also has rounded leaves.

Sweetgums aren't going to develop a low canopy for climbing though. But the fall color is excellent. They're difficult and slow to transplant but heat, drought, poor & poorly draining soil tolerant. In most of florida sweetgums color well and hold the colored leaves for practically the whole winter.

Sycamore, fast growth not weak wooded nice bark can live 500 yrs according to my feild guide. If there's no aversion,even a replacement live oak.That way you'll have color in winter even if it's green.Live oak, your original tree is supposed to live very long too,I wonder what it died from.

Swamp Chestnut Oak (Quercus michauxii). Large, sturdy, fast grower fo an oak, and it has good fall color, too.