I grew up in Florida, and we've been going back most years since my teens were born (cancelled 2020, but have just booked for Oct 2021, fingers crossed!!).
Florida is about the size of the UK, but driving around it is absolutely fine. However, you should give some thought as to how much time you want to be sitting in a car! Miami to the Keys would take a few hours, and I think you should really take the time to stop on the way, the drive is spectacular, but Key West is very different to the other Keys in feel. The Keys don't really have beaches as such, be prepared.
Miami is a lot of fun, you have the city itself as well as Miami Beach (which is a different city). On Miami Beach you've got the amazing Art Deco district, and the beach is beautiful and quite iconic.
You can choose between the Gulf Coast beaches or Atlantic Coast beaches. My favourite is Gulf side (Naples, Marco Island, Sarasota, Siesta Key, Anna Maria, Sanibel & Captivate...), but there's lots to be said for Atlantic too (Palm Beach, Daytona, as well as Miami etc).
Everglades worth a visit, I love seeing the alligators!
Further north, St Augustine is (I think) the oldest continually occupied city in the US, very touristy, but I really like it. I've not done J'ville or the Panhandle, although it's on my list!!
We drove Miami to Savannah in Georgia last time we went, took about 7 hours of driving in total (took advantage of jet lag and left at crack of dawn, stopped in St Augustine for lunch, got to Savannah late afternoon), if you fancy something a bit different. It was beautiful. Would love to do Charleston too one day.
Weather wise, I'd avoid August and the summer months, the humidity is absolutely disgusting, and you'll get eaten alive by mosquitoes. Plus you'll get caught up in tropical rainfall, potentially hurricane type weather (but you'd be unlucky for that extreme!). Personally I'd avoid any later than May and any earlier than end September. We usually go at Easter, and enjoy 25-30 degree non-humid sunshine (depending how far south you are as well).