Raising your skills with weapons? Options

[Count Valerius]

I see people in the fighters guild+arena and stuff hammering away at dummies with their weapons. Also i see targets for practicing archery in some places, like by the fighters guild. can you hit these with your weapons or shoot your bow at the targets to increase your weapons skills?

[Gomez']

I leveled my marksman entirely by shooting creatures in the countryside.

My suggestion to quickly beefing up for any respective quest or random encounter you might have.

Use your spare time and wander around the province aimlessly.

I leveled like 4 times on a couple round trips to the edges of Cyrodil, just by slaying Goblins and murdering bandits, raiding caves', wandering abondoned elvish temples and such, stopping in towns to restock on munitions and repair armor and weapons.