Grips on the Downtown are standard motorcycle diameter. Don't get the one for cruiser motorcycles (too fat) nor the ones for ATVs (don't want both sides to be same size).
To replace grips, you basically wedge a screwdriver between the handlebar and the rubber and try to break the seal. You can use soapy water or light oil to help you. Sometimes compressed air will take the grip off easily. The throttle side is a little trickier, as you have to get the rubber off the nylon throttle sleeve.
If you don't care about saving the old grips, then do it the easy way and take a box cutter to slit the grip lengthwise. Peel off.
Slip the new grips on. I don't use glue when reinstalling the grips, as the sticky rubber won't move. Glue also makes future removal much more difficult.
As for tires, try the Michelin Pilot Pure SC if you cannot find City Grips.