Yep. Those are stinkin' awesome cars.
The biggest problem those things have is the headgasket - but they don't fail on their own. The cooling systems can be somewhat wonky on these, there are a lot of plastic bits that like to crack and crumble, and when these things overheat and pop the headgasket they typically pull the head studs. This is because the studs expand at a different rate than the cylinder head and engine block, and what happens is the threads get pulled out of the block. You can heli-coil them with the right equipment, but it usually doesn't last more than a year or so; don't get me wrong, occasionally when you pop a headgasket it won't pull the studs, but it happens more often than not. So that's the biggest thing to be aware of.
Check the cooling system out, all the hoses and elbows, for leaks or spots that have dryrotted/cracked/deteriorated. The most common leaks on those are from the coolant expansion tank itself, which usually cracks. Pop the expansion tank cap and check the coolant level - it shouldn't be all the way to the top of the tank, but you should be able to see it. There's a little red rod that sticks up out of the tank to tell you the level, for the most part it should shoot all the way up - there's a little diagram showing you the level the rod should sit at to show you the level.
Also on top of the coolant tank is a coolant bleeder screw, that's used for bleeding the coolant. Make sure that isn't leaking or cracked.
Another big thing on these is people improperly bleeding the coolant - there are a few nifty steps you need to take to properly bleed it. Improper bleeding can cause air pockets and overheating, but the most easily noticeable thing is that the heater won't work. My usual first step in looking one of these cars over is, after I've verified coolant level, to start the car and turn on the heater, and let the car run while I'm looking everything over. If the coolant level is proper and the car starts/runs well and the heater works, the majority of the 'major' problems those cars typically run into are probably nonexistent.
Beyond that, the power steering system is prone to leaking - there's an o-ring on top of the power steering reservoir that ALWAYS leaks, and costs only a few dollars; the hose clamps on the power steering lines tend to leak as well, but if that's the case the fix is simply to put new screw-type hose clamps on. Occasionally the hose that clips onto the power steering cooler leaks, and if that happens the hose just has to be replaced; most common thing is the hose clamps though.
Valve covers, front seals, oil pan gaskets, and rear main seals all like to leak on these; the rear main requires pulling the transmission, and the oil pan requires a decent bit of work, but these are typically good bargaining chips on a used BMW. More often than not, they're going to at least seep.
You've got basic maintenance items like spark plugs, air filter, engine oil, etc that all need to be in good shape. Castrol synthetic 5w30 is what these things require, when I don't use Castrol or Mobil 1 (which, I don't usually use Mobil 1 in BMW's), I use Oreilly auto parts house brand synthetic 5w30. The stuff is great. For engine coolant, ONLY USE OEM BMW COOLANT. Blue stuff. Don't use anything in the cooling system EVER other than OEM BMW or Pentofrost (which, you probably have never heard of and don't know where to get);
There are a few vacuum lines that like to dryrot and fall apart on these, one comes all the way around the valve cover along the passenger side up to the secondary air injection valve - make sure that line isn't broken or falling apart. There are a few other lines on the intake manifold side, check those as well. The secondary air injection pump bushings on these usually break, so the pump is probably flopping around; when you first turn the vehicle on, you should hear the pump come on loud as shit for a few seconds when the engine is first running; it's not uncommon for the pumps to eventually fail after flopping around on broken bushings though. If the pump fails though, it'll throw a check engine light.
That's another thing, be sure and get a handheld generic code reader - even a cheap one from AutoZone. You NEED to check the codes on this thing, whether or not the engine light is on. Any codes it has, just ask and I'll address those as necessary.
Those have one of two transmissions, either a ZF or a GM. Both transmissions are 'fill from the bottom' style, as most BMW's are. If/when it comes time for you to DIY a transmission flush, I'll walk you through that. There's not really an easy way to check the fluid in the transmission on those, so just make sure there aren't any leaks and that it runs and shifts properly. Those transmissions are pretty robust and don't have many problems, and when they do it's because of low fluid or poor fluid quality. FYI, it's very uncommon to rebuild those transmissions, as the parts necessary to rebuild cost just as much as another transmission. Unless it's a manual transmission, which means that the person who bought it new must have thought that it was 1980.
Suspension bushings on these in the front wear out pretty regularly; the rear isn't anything that I've ever seen any major issues with other than sway bar end links. RockAuto is going to be your best bet on getting replacement suspension parts - there's a triangular shaped lower control arm with two ball-type joints built in, and both of those wear out over time; the bushing that the rear of the control arm mounts in is also notorious for wearing out. Inner and outer tie rods are another big suspension wear item - however all of these parts are relatively inexpensive from places like eBay or RockAuto, and not too terribly complicated to swap out yourself. For suspension components, I have not seen a noticeable difference in wear and usable life from the eBay/RockAuto cheap brands versus OEM or Meyle stuff (I think you may be able to get Meyle from RockAuto, actually)...Dorman makes good quality suspension components for these, as well.
Window regulators on these are junk, as well. If you use your windows a lot, expect to replace at least one regulator a year. You can get cheap window regulators or you can get ridiculously expensive ones from the dealer, and both of them are going to last the same amount of time. They'll start going out and popping or being slow to roll up or down, or they just won't - completely normal, the regulator design on these is junk.
Back to the engine, another common maintenance item I forgot to mention earlier is the crankcase ventilation control valve, which people call all sorts of weird things (CVCC, CCCV, PCV, Swirl valve, cyclone valve, etc). These have a diaphragm in them that likes to tear, and/or they'll leak oil. It's under the intake manifold, and is a hundred bucks or so with all the necessary hoses to replace it; not too terribly much trouble to replace, although if you have to do one and you've never done it before it's best to remove the entire intake manifold so that you have a better view of what's going on.
Off the top of my head at 5:30am, that's the majority of the problems those cars have...the E46's don't tend to have too many electrical problems, they're really damn reliable cars overall. They have maintenance reminders build in - when you first turn the key on, the little green/yellow/red bars on the instrument cluster that probably say "Inspection" show the oil life, and the time left until you need an Inspectin 1 or Inspection 2 - these are the two BMW services. These don't take 'regular oil changes', and you shouldn't just change the oil every X amount of miles and put a sticker on the windshield. Inspection 1 is basically an oil change, Inspection 2 is an oil change/diff service/all air filters/spark plugs/fuel filter deal. Hopefully the person selling it doesn't have a burger rack sticker on the windshield, and hopefully they were smart enough to at least reset the maintenance indicator - it's reset via the buttons on the cluster, I can talk you through that if you're too lazy to Google. Also, the brakes have wear indicators, and throw a brake lining warning light on the cluster when they're low. A common thing though is for people to replace the pads and sensors (you HAVE to replace the sensors when you replace the pads, or the light won't reset) and not properly reset the warning indicator - so it's very common for a car with new brake pads and sensors to still have the warning light on. If that's the case, to turn that off you need to turn the ignition key all the way on without starting the car and walk away for 30 seconds or so. That's all it takes to reset the brake wear sensors, yet there doesn't seem to be a fucking shop in Texas that knows that. Harumph.
Uhh....yeah man, that's pretty much all I can think of. Obviously look out for shit like body damage, rust, obviously broken shit, etc...but, remember oil leaks are common (if not normal), and are inexpensive to fix (but not necessarily easy), and ultimately make great bargaining chips for negotiating down the price on these cars. Can't think of anything else off the top of my head, I'll revisit this later and see if I can think of anything else.
Oh, also, the headlight lenses are probably all clouded up - you can get replacement lenses pretty cheap, and they're wicked easy to swap out.