First, ally with everyone except the Retros (while destroying 6529 Retro ships is a great way to ally with everyone, doing so takes forever in a Centurion and I don't want to even *imagine* what it would be like with the Tarsus). You can do this by destroying mostly Retro ships during the early part of the game. It will make the Merchants, Hunters, Confed, Militia and Pirates like you. Try to avoid destroying ships that belong to other factions. Once you've destroyed 25 Retro ships, the Pirates will become neutral assuming you haven't destroyed any pirate ships. To ally with the Kilrathi, simply build up your reputation with Confed by destroying lots of White Talons (i.e, Retro ships), then head to Perry Naval Base. Land. If you are using missile launchers, sell them and buy torpedo launchers and as many torpedoes as possible (20). Take off. If you don't see a Paradigm around, land and take off again. Do this until a Paradigm spawns near the naval base. There will be 3 Stilettos guarding it. Ignore them and quickly destroy the Paradigm with your torpedoes. The Confed ships will barely have time to realize what's happening, and before they can react, the Paradigm will be destroyed. Quickly land before the Stilettos rip you apart. Repeat until you get your Cat reputation to an acceptable level (assuming you haven't destroyed any Kilrathi ship, destroying a single Paradigm is enough to make the Kilrathi neutral towards you).
Second, when the odds aren't in your favor, consider luring the enemy into asteroid fields, or places where you can find allies (sometimes you can do both at the same time). If there's an enemy in front of you, and you're inside an asteroid field, try to destroy the enemy vessel with the asteroids. Simply move towards it and dodge any asteroids that are headed your way. Use your afterburners wisely to make this trick even more efficient (the faster you move, the more asteroids are generated).
Third, don't be afraid to mess around with your shield settings. I recommend getting Level 2 Shields, but you can use the "S" button to lower your shielding a bit if you need extra energy to run away. One trick is to lower your shield level, then use Shift-S to raise it immediately afterwards. You will get extra energy until your shields are back to full.
Mission-specific advice :
Roman Lynch Regis mission : Fleeing from one Stiletto is one thing, fleeing from 3-4 at the same time is another. I'd recommend destroying the first two Stilettos that attack you, then you can just fly away to your destination while avoiding the last Stiletto's blasts. Not destroying the third Stiletto makes sure that the second wave doesn't spawn.
Masterson : The Demons are pretty tough but they're nothing you can't handle with a little in-system help from the Pirates. The escort missions are easy enough once you know which ships to get rid of first. In my opinion, the Black Rhombus isn't that bad, either.
Lynn Murphy : This is where the "fun" really starts. For the first two missions, lure the enemy towards the Palan asteroid field. I'd recommend getting some help from the Pirates that sometimes spawn there. You may have to retry these missions a few times as they're somewhat difficult, but the asteroids really do make things much easier. Also, ships destroyed by asteroids or allies do not affect your reputation so that will prevent the Bounty Hunters from getting too mad at you. As for the third mission, getting this far means that you can ignore the blockade ships and just land on the planet to get Dr. Monkhouse, thus "skipping" the mission altogether. However, Lynn Murphy won't pay you if you do this; so if you really want to beat this mission the normal way, start by destroying the first allied Brown-Winged Talon that spawns near the blockade. This is necessary to avoid a game-crashing bug (in fact, you may need to destroy both of these Militia Talons). Then lure the enemy into the asteroid field as usual.
Taryn Cross : Err, yeah, Mission 2 and Mission 4 aren't a problem at all if you're allied with the Pirates and Kilrathi, but Mission 1 is hard and Mission 3 can be hellishly difficult. Again, try to use the asteroid fields to your advantage and don't be afraid to hit the hardest Navpoints first. Remember that leaving an enemy ship alive can mean that the next wave won't spawn. Trying to destroy the Kamekh may not be a good idea, either.
Steltek Drone chase : Perhaps the hardest part of the game. Dodging the drone's shots can get ridiculously difficult and outrunning the Drone will likely only be possible if nearby ships are willing to distract it. Firing a missile at the drone will also distract it for a bit, but this particular trick doesn't work that well in a Tarsus. You may need to get a bit lucky here.
Final Showdown : Depending on how lucky you are, this may seem very easy or very difficult. The Steltek Drone can be difficult to hit. Sometimes, the missiles that you and your allies will fire at the Drone will distract it long enough that you'll be able to destroy the Drone while it's doing its evasive maneuvers (possibly with a little help from ITTS). And sometimes, it won't be enough and you'll have to face the Drone head-on. Fire a few shots while the drone is headed towards you, and dodge to the best of your ability. Repeat until it dies. This may take a few tries but eventually, you should be able to destroy it.
Should you succeed, I'd suggest using the Boosted Steltek Gun during the generic postgame missions to earn lots of credits so that you can buy the Righteous Fire upgrades. The Priv Tarsus may be a joke, but a fully upgraded RF Tarsus is much closer to "ruthless killing machine" than "piece of junk". In fact, you probably won't have any real trouble until the very last missions.
If you want an even tougher challenge, you can try finishing Privateer with 0 Kills on your Killboard. The only ship that you need to destroy personally is the Steltek Drone, and it doesn't show up on the killboard. Note that this particular challenge may require an insane amount of reloads (and do note that I've yet to beat Righteous Fire that way - I don't even know if it's possible, but quite frankly, after finishing this pacifist run I didn't really feel like even trying to do the same thing in RF).