I'm trying to put this into perspective for you. Porto to Lisbon is about the same as Bethesda to Newark, with Coimbra relatively in the same position as Philadelphia.
You can see train schedules on the official site. Note that the main train station in Porto is Porto-Campanha and in Coimbra is Coimbra-B. You will want to take a taxi to/from the train stations to the town centers. The main train station in Lisbon is Oriente and is linked by metro to the historic areas. For any iteration of the trip you're talking about, a car (or 2?) would only make you miserable and waste a lot of your time and money.
If you can fly open-jaw, you could spend the bulk of your time in Lisbon with day trips and then Porto. If you have to fly R/T Lisbon, it may not be worthwhile to include both cities, since you'll lose too much time in travel, even if you have 10 days. If you don't have more than a week, I don't think it's worth it to add Porto - you'll lose the better part of a day just moving to a new location.
I had a car and spent 2 wonderful nights on the Douro River at a winery/inn, so I don't know how you'd get there, maybe an organized tour or a river cruise. An easier day trip than Coimbra is Aveiro, which I enjoyed, though perhaps not as interesting as Coimbra.
There are a lot more options for day trips from Lisbon and much more to see and do than in Porto - though I loved wandering in Porto more than in Lisbon.