A Blaster Won't Do Much: Level Control

Level control is what separates the good from the best. It doesn't matter how good your railgun aim is, or how well you can lead rockets; if your opponent doesn't let you get the railgun or rocket launcher you're not going to get very far at all. There are two main types of level control: Weapon Sweeping and Spot Control. Weapon Sweeping is most effective on small levels, whilst spot control is more suited to larger levels where there is a depot of health, armour and ammo.

Weapon Sweeping
On small levels, weapon sweeping is very effective. It involves running around the entire level picking up all the 'good' weapons, then timing it so you begin the circuit just before the first weapon respawns (this is, of course, assuming you are playing with weapons stay off - as you should be in a 1-on-1 match).

To weapon sweep successfully you must have good timing skills - each weapon takes 30 seconds to respawn; so ideally you want to finish the circuit in about 28 seconds. Of course in reality you're going to have interruptions - like killing your opponent, for example ;) - so it's best to give about 20-24 seconds for a circuit. Any more and you either want to make sure you meet your opponent near the end of your circuit; or use spot control instead.

Q2DM3 is a level where weapon sweeping is possible, however it can be risky as it involves a trek through the frag pipe. Start from the rocket launcher, drop down to the rocket pack and run around the top to the railgun. Drop down and grab the hyperblaster then continue along and climb up the ladder to the super shotgun, then jump across to the armour and then up to the hole leading to the rocket launcher again. Note to make the jump across to the hole from the yellow around you need to employ a trick known as a strafejump; described in detail in the next section. This route should take about 20 seconds, leaving you with time to rocketjump to the invulnerability (which usually isn't there as most 1-on-1's are played without powerups). Whilst it doesn't yield the powerup it does give you back 75 health and a railgun; so it's useful if you have less than 100 health and little armour. However, for the Q2DM3 situation I much prefer the Spot Control method.

Spot Control
No, this isn't putting your dog on a leash - spot control involves selecting a nice warm, cosy area of the map and controlling it. The area selected should contain health, armour and ammo - so you can move out at intervals, take a few shots, then dart back in to stock back up on supplies. This, of course, has the downside of letting your opponent go out and grab weapons - and armour - so it's important that you kill your opponent before they have a chance to stock up.
Note that spot control is not camping. Camping is positioning yourself so there's only one entrance to your hidey-hole, and placing your crosshair just before a major weapon or item like a BFG, rocket launcher or red armour. Usually the railgun or rocket launcher is used for camping. Spot control is just making sure no-one can access your supplies, whilst still moving out of your area to attack. Basically, you're making sure you can get back to the 'spot' before your opponent can.
Spot control in Q2DM1 is important - you want to control the megahealth room, as it contains 120 health, 50 armour, a backpack and 200 bullets (including the chaingun). From here, you can also hear your opponent anywhere on the map - for example, you know your opponent has just picked up a rocket launcher because either:

a) you saw them head for it in the Central area (as you should always be looking out there),
b) you heard them drop into the lower rocket launcher room,
c) you heard them pick up armour shards, or
d) you heard them rocketjump (rocketjump is the only way to pick up the upper rocket launcher without touching the armour shards).

You can also hear whenever your opponent is about to enter the megahealth room - either:
a) you saw them run up from the super shotgun (as stated above, you should always be looking out here),
b) you heard them use the lift or rocketjump (meaning they're about to drop in near the chaingun),
c) you heard them get out of water - this means they either just got the railgun, or just came out of the machinegun tunnel - and can either enter to megahealth room, or go into the machinegun tunnel (in which case you'll usually hear them enter water), or
d) you heard them jump to your right, pick up ammo to your right or fall to your right - those three bullet packs under the upper rocket launcher serve a very important purpose on the map. Your opponent cannot sneak up on you with them there.

Also, in the megahealth room you are relatively high up - this means you have the advantage when attacking your opponent from above.
Q2DM3 also benefits from spot control - the area you want to control is the yellow armour room. Here you have quick access to 20 health, 50 armour, shells, slugs, cells, and you can hear your opponent grab the rocket launcher (in which case it's a nice easy rail).