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I have a Minmitar Slasher outfitted with a very good afterburner which I use for running delivery missions. I can get it up to almost 1,000m/s even in combat. I was forced to use it on a combat mission with about 10 pirates. I didn't expect a frigate to be a match for the enemies (which were pretty weak) but I gave it a shot anyway. I found that I could stay continually out of their range since my ship is way too fast to catch. I hammered them with my light railguns from about 10km away and only took a couple hits throughout the whole battle.   

My guns can only hit enemies at about 10km range right now. What do I need to do to boost this range? I think if I could get high enough range in that ship, I could keep far enough away that most rats wouldn't even get a shot off at me. Any thoughts? 

You could train the Sharpshooter Skill, which increases your optimal range by 5%/level. You can also use Iridium Charges as they will increase your range by another ~20%.

If you like that kind of combat, you may want to look at assault frigates. For example, the caldari harpy assault frigate (will require caldari frig 5 on top of what you currently have) gets 2 10% bonuses per level to small railgun optimal range, meaning it can hit at massive range (around 100km properly fitted) with small guns! The gallente Enyo also gets one 10% bonus per level to optimal, along with bonuses to damage, tracking and resistances, and may be more your thing.

 Aside from that, Narkes is correct. The sharpshooter skill will increase your optimal range (there is a skill for falloff too, but that requires gunnery 5 to learn). Also, there are other ammo types which have greater range than iridium. I think iron is the longest, at 60% bonus. If you use tech 2 guns, you can fit 'spike' advanced railgun ammo, which gets an 80% range bonus, at the expense of tracking, and does more damage than standard iron.

 Another thing which may help is fitting tracking enhancers, a low slot item (actually boost optimal much more than tracking).

 Of course you shouldn't really be using hybrids on a minmatar ship, but I won't go into that =P (anyway, it's probably not worth it for you to train the skills, unless you plan to do this often).
 

Wow, I think I might have to look into this sometime when I need a break from the painfully slow ships I normally fly. It is so much fun to zoom around out of range of your enemies. Of course it will be a while before I leave the peaceful life of a miner and get into the glamorous combat pilot lifestyle.  Now if I boost optimal range, it doesn't reduce my accuracy at closer than optimal ranges, right?

If I'm not mistaken, your falloff  determines your minimum and maximum range. If you have a weapon with for example 14km optimal and 3km falloff, your shots will hit at a range of 14 +/- 3km. (at least range wise. tracking speed not considered)

When you increase your opt range to say 16km, it would automatically raise your minimum range to 13km.

I'm not sure about this though.

 

As far I'm aware, there is no minimum range. Optimal is the max range at which you still have a 100% chance to hit (ignoring all other factors, i.e. tracking, signature radius, etc). Optimal + 2x Falloff is your absolute max, outside that you have a 0% hit chance. At Optimal + Falloff, you have a 50% chance.

 However, the graph is not linear (it's like an S, I forget the name). at Optimal + 1/2 Falloff, you still get about an 85% chance to hit.

 
Your minimum range is only really determined by your tracking speed. Once the angular velocity (a useful column to add to your overview!) of the enemy exceeds your weapon's tracking speed, you won't be able to hit them. Obviously, a very close enemy usually has a very high angular velocity, while a distant enemy has a very low one.
 

That's an interesting twist to the whole idea. If I were to fly right up into someone's face, and start orbiting their ship at say, 1,000m/s, would I most likely exceed their tracking speed and be able to get out without damage?

Also how do drones factor into this? I'd hate to be able to avoid damage from a battleship, only to get shot down by a puny drone. There are so many factors to consider in EVE Online, it's downright scary.  

What you're suggesting is exactly the principle behind 'speed tanking' (basically maximising your angular velocity against a target to avoid being hit), one of the dominant forms of pvp tanking for cruisers and below.

 Drones also have a tracking speed, but that is much faster than most actual ships (thus drones are a battleship's primary weapon against small fast ships). However, if the ship can outrun the drones (not too hard vs t1 heavies, and some ships i.e. interceptors can even outrun warrior IIs (the fastest drone). This is the main attraction of the minmatar drones (warriors in particular).

What I'm not sure about is whether a drone has to slow down to it's orbit speed to attack, or if it can shoot at it's mwd speed.