How do I properly use the Adjudicator?
Immediately put crew people at pilot (duh), weapons, and shields. Since Zoltans give 1 power to each system, you get 2 bonus power doing this. Pay for one reactor upgrade, and bam you can now power the Halberd beam.
But how do you get through people's shields? The Halberd beam is a 2-damage beam. It will pierce through a 1 point shield, losing some of its strength but still doing considerable damage. For now, that's what you do. You can alternately fire the Leto missile once and then switch to the beam, but I find it's frequently not necessary.
It behooves you to get a good weapon upgrade asap. Any thing that is good for either pounding on shields or reducing shield capacity. Another point in weapons allows you to use the Leto, which you should only break out on 2-strength shield opponents. Optimally, you want a good energy weapon(preferably a blaster Mk I or II, or an ion weapon), or a more powerful ammo-using weapon like the small bomb (2x the damage for the same power requirements as the Leto, the small bomb is an exceptional option). Drones are certainly an option; I like having another energy weapon as a mainstay because they don't cost ammo or parts. Once 2-shield enemies are showing up the halberd beam needs help.
The Halberd beam can remain a mainstay for the rest of the game. Excellent weapon, as long as you can tear down shields.
Zoltan crewmbers are not good at fighting. You'll need to find proper boarders if you get a teleporter.
A new crew-member should be a priority though so they can take over as pilot. Move the pilot-zoltan to the engines. That gives you 3-free power, which is quite an advantage.
Disclaimer: I haven't played on Normal yet, only Easy.
Put your starting crew where they power something that frees up energy (weapons, engine, shields, etc.). You need one to pilot when you want to jump, then you can move the pilot to shields/weapons/engine to get extra energy for battles.
You can also unpower your medbay for extra energy, so that you don't have to move your pilot (but doing both lets you use both your weapons after your first weapons upgrade).
On Easy, you can pretty much two-hit any ship in the first and second sectors with just the Halberd Beam, before your Zoltan Shield runs out, even. Pause and take your time to place the Halberd Beam so that it hits four or five rooms -- it's a hit even if the red line is only one pixel past the wall of the room. I've only seen one ship where only three rooms are reachable, and it's a weak, shieldless auto-scout.
If you place your Halberd Beam lasers so that it goes through the shields room first and it takes down their shields, you can do full damage on all the other rooms. Of course, if their shields come back up before your lasers hit all the rooms, you'll do less damage.
Hope this helps.
All the above are useful as a general guide. Below are some "mastery/achievement tips".
One thing I've noticed is that you can depower the shields until the Zoltan shield is down. This frees up yet another energy point for weapons or dodge. Personally, I put it into dodge, as it means when you do need the extra energy back to the shield, you don't risk resetting the cooldown.
Also, Ion does double damage on the Zoltan Shield, so dodge is 100% your best friend.
Finally, if you're going for the "No Extra Power till Sector 5" Achievement, you wind up saving tons of scrap (as there's no point buying extra slots). Buy cloaking ASAP. It's 300% more effective than the equivalent power going into the engines, and it keeps your Zoltan shield up longer (see the first point for why this is a very good thing). Another useful feature is level 2 Oxygen, as it means you can spend lots of time with it completely unpowered, then burst it on full for a few seconds when you get near suffocation.
Obviously, for the 29 power in systems, you need to be on the lookout for another Zoltan crew member. They're rarely handed out as event rewards, so always have enough scrap available for them at a store. Also, avoid the "Great Eye", "Spiders", "Mantis Pod" and "Dead Station" events, as they kill random crew-members. You also need to focus putting money into systems to get up to 29 slots there.