In my opinion the best time to visit Cairo is either the fall (october to mid-november) and spring (mid-march to april).
Places to visit depends on minaly 3 things: budget, amount of time you're planning to spend in Egypt and your interests.

If its Egyptology and ancient Egyptian stuff you're interested in, then it'd probably be Cairo, Luxor and Aswan (with a half day trip to AbuSimbel as one of the previous posters recommended, I find the story of moving the temple very impressive).

If its scuba diving you're interested in, start by Cairo (as you want to see the Pyramids) and then do some diving in the Red Sea. My favourite 2 spots are Sharm El Sheikh and Dahab. You can opt to stay at one of the resorts there, or if you're a serious diver you could consider taking a safari boat for several days.

As one of the posters recommended, Alexandria is 2-3 hour train ride from Cairo, interesting place, make sure you see the Alexandria library, Roman amphitheater, the catacombs, qayt bay fort, etc.

One other thing to note, Egypt doesn't only offer ancient Egyptian stuff, it has things from other more recent eras too.

Have a lovely trip and tell us what happens

Cheers.