Let's say you go with d6's for movement
How do you differentiate between people who move at normal speed, slow pokes, and speed demons?
Normal speed: Roll 1d6, that's how many squares you move
Slow poke: Roll 2d6 and take the lower result
Speed demon: Roll 2d6 and take the higher result
If you've decided to stick with 6-siders, and don't want the max move to exceed 6
that might help.
As far as "what should I use for attack/defense/the entire rest of the game"
Well
If you've taken a stand that 6 siders are the easiest to find, then you may as well stick with them. There's a huge list of options when it comes to using a d6
The GW method:
Seven minus some stat number (1-6) is the result you must match or beat on a d6 (adding applicable modifiers). You can test large groups in this way, since each individual die represents an attempt.
The Bell Curve Method
Using the results of 3d6, try to exceed a certain number. The probablility is a lot more granular, and dice rolls must be done one at a time to avoid confusion
The Craps Way
using 2d6, try to get a certain desirable set of numbers. If you fail, give all your money to someone named "Larry".
So, depending on how many dice are "firing" to resolve a combat, you'll want to use die singularly, or in pairs or triplets.
Do you have an idea of how many figures you intend to field, and whether each of them will want separate tests, or if they'll fight as a mob?