The first thing to do is get a gauge/monitor solution. You need to see what's going on with the motor before you start blowing money on stuff that won't make it reliable. Some of the key parameters will be ECT, EOT, Fuel psi, FICM, just to name a few. Read this link to better understand why.
Things to consider about owning a 6.0L
While your making your decision on a monitoring solution, get a full Oasis Report from a Ford Dealer, this will tell all the work that has been done to your truck when it was brought in for warranty work.
After you have a monitor solution, and your ready to dig into the motor to fix any known issues, your gonna want to update certain parts of the motor that have issues per the year motor you have. The '05's have issues with the Snap To Connect (STC) fitting, there is a updated part for this, along with dummy plugs in the HPOP rail. Fuel Pressure Regulator (FPR) spring, the original spring gets weak and causes low fuel psi that can/will cause injector damage.
Now, to delete the EGR cooler or not. It's illegal to delete an emissions equipment per EPA. BUT! a lot of us do it anyway. Now this will also depend on what state you live in, some states are stricter than others when it comes to looking for emissions equipment. If your in one of them states, look into Bullet Proof Diesel's EGR cooler to keep you legal. Some day the EPA might reinforce the laws, and those of us who are delete will have to put on our emissions equipment.
Aftermarket exhaust- you can go a couple of ways here. A larger OD exhaust and maintain the CAT and aftermarket muffler (CAT back system). There is also a turbo back system that maintains the CAT. You can also go with a straight pipe (off-road) system, no cat or muffler. It's a matter of how much money you want to spend on the size OD of the pipes, and what the pipes are made out of (stainless, aluminized, steel).
Aftermarket air filters- Cold Air Intakes (CAI) versus the stock air filter. Your not gonna find a better filter for filtration than the stock filter. Not only that, but the stock filter can handle up to 500hp of air flow. Most people put on a CAI to save money in the long run, and the ease of cleaning/replacement. Bling is another reason for a CAI. Air cleaners are a big debate, if I can find a stock set-up for cheap, I'm going back to it. There is no real HP gain from a CAI. If your willing to take a chance with a CAI, you might end up dusting the compressor wheel of the turbo, and having to replace it, so how much did you save with a CAI.
Now onto the performance talk. With the 6.0, you'll want a programmer that is capable of custom tunes that also reprograms the TCM (tran control module). Most go with a SCT device, but there are some others out there like Power Hungry Performance (PHP) Gryphon. What ever you do, don't go with any programmer that has shift on the fly tuning, they don't reprogram the trans enough to handle the extra power that the motor will produce.
As stated by PGreenSVT in the link provided, custom tune writers aren't gonna wrote tunes that will blow your head gaskets. If they did, you would be reading bad reviews on the powerstroke forums. When you are ready for custom tunes, tell the tune writer that your not studded, so they can write you a tune that is safer for your truck. Now just remember, things are bound to happen when you tune, your putting more stress on the motor, so tunes are gonna find the weakest link.
Maintenance is the key to a healthy motor, using a synthetic diesel oil will make your injectors last longer, along with being anal about oil changes in a timely manor. Clean oil will be your best bet for happy injectors.
The Torqueshift trans is pretty strong, again be anal with fluid maintenance.
The powerstroke forums will be your best source of info, read as much as your brain can handle in order to make the best decision for your truck. Over the years, the 6.0 has gotten a bad rap, but with forums like the org and aftermarket companies, the issues can be resolved.
Here are a couple of links to get you started, the first one is the 6.0 info manuals. Start with the '03 and read up thru the years in order to get an idea of sensor location and what the parts are.
This next one is about bulletproofing a 6.0.
Write Up: Bullet-Proofing the 6.0L PSD.
Now I can sit here for hours providing links, but I don't have the time. Like I said, start reading up on the .Org. You'll catch on.