No, convertibles don't have handles. I would definitely recommend an infant seat in the beginning - way easier.
I don't know about Graco stroller capabilities but most stroller brands at this point have some sort of car seat adaptor that can be used to put the car seat into the stroller.
after they are done with an infant seat, is convertible the next seat they go into?
Yes, and there are different options for convertible car seats as well. FI and I both just bought new carseats for DS. There are convertible car seats that only go up to a certain weight/height limit and then you have to buy a high back booster, then just a regular booster.
We both bought a carseat that is a 3 in 1, meaning that it is a convertible car seat, then the harness can be removed for a high back booster, and then the back can be removed to become just the booster seat. DS will use it, and LO will use the seat as a convertible seat when he has grown out of his infant car seat.
While some people don't like to lug around the infant car seat, I definitely didn't mind. It was easy to not have to risk waking a sleeping infant and just throwing a blanket over the seat. Some people prefer to just go right to a convertible that fits infants from 8+ pounds so they don't waste money on the infant car seat. It's up to you and your preference, though.