Rhodes is one of my favourite places in the world.
Lindos is gorgeous, and a must-see. It's very safe, and very pretty, get's horrendously busy and crowded in high season though. This is where my single mum used to take us for holidays when I was a teenager, as it was safe and small enough to allow me some independence.
Pefkos, near Lindos, is a bit quieter abd doesn't attract as many day trippers. It has lovely beaches and is a good place to base yourself.
Faliraki was the big "party" resort. I personally wouldn't want to stay there with kids, but it's where most of the water parks are.
Rhodes Town, especially the old part, is gorgeous and good to visit. Not as "beachy" as other parts of the island though.
There are buses, though they are slow, hot and crowded (or used to be - I haven't been back in a while, maybe they've improved). There was always plenty of taxis, but the driving of them can be a little...erratic. The many roadside shrines are unnerving as you whizz past them at high speed whilst your very friendly driver turns round in his seat to chat to you!
It does get hot as hell there in the summer - Lindos particularly, as due to a quirk in its geography it's usually about 10 degrees hotter than the rest of the island! I remember our first visit there was in August, and our first night we ended up swimming across St Paul's Bay at sunrise just to cool off!
I'm sure you'll have a wonderful time if you go. I'm "saving" Lindos for when my two are teens, so they can have a similar fun introduction to it that I did. 