I love my eBags backpack. I've used the MotherLode weekender for years and adore the dang thing. It's carryon size (as long as you don't expand it) but holds way more than it should be able to. Do be careful not to overload it (very easy to do) since you'll be carrying it.

A simple carryon size roller would probably work fine too. You don't have to roll a huge bag, carryon size is plenty even for 3 weeks.

Just depends what you personally want. There are 'convertables' which are backpacks with wheels that weigh a bit more. Regular lightweight nylon backpacks. If I'm not using wheels (a 22" roller) personally I prefer a duffle bag with a good shoulder strap myself. I find that more comfortable to carry for city/train travels than a backpack style.

I love my Osprey Porter 46; it has a good waistbelt which helps transfer the weight off my shoulders and onto my hips when I'm wearing it as a backpack, and it can be carried by it's handle or with a shoulder strap as well. And Osprey has a great warranty on their packs, too.

Whatever backpack you end up with, be careful if you plan on flying any of those low-cost European airlines like Ryan Air. Many backpacks designed to meet US carryon sizes are a smidgen too big to be legal carryons on those low cost airlines.

I know how hard it is to choose a bag because when I first started my long-term travel, it was like a chicken and egg problem.

Should I get a bag first and try to fit everything in?
Or should I lay out everything I want to bring, then find a bag?

The latter would be best since you sound like you want to travel light.

Here's a good one: http://www.deuter.com/en_US/backpack-details.php?category=119&artnr=34211&title=Futura%2028