My inclination is to do a teardown, cleaning as you go and checking every part. Besides giving you the opportunity to do a deep detail on the bike it will make you intimately familiar with the your bike and what makes it work as well as making it as reliable as possible. With a month or so until good riding weather, it's a good time to do it.
I like buying my parts from a couple different sources. MaxBMW is great, they have a nice online parts list and you can order online with free shipping on orders over $100 and they don't charge sales tax.
Universal parts like fuel lines, filters, fluids, etc., I get locally at places like NAPA and Walmart. Brakes, tires and batteries I get online, eBay has good prices. There are also aftermarket sources; EuroMotoElectrics is a good one as well as some of the European sellers. Personally, I have never been impressed by Beemer Boneyard, but YMMV.
eBay is also a good source for used parts. By scanning the listings periodically you can see what the current market is for various items. I have had good luck with budget brake pads and tires on eBay. It's also a very good source for used body parts. There are a number of bike breakers who sell a wide assortment of parts there.
About your bike, get your rubber bits from MaxBMW, fuel line can be had at the local NAPA as well as the filters for fuel and oil. Air filters can be had for a good price on eBAy. Get your oil(Castrol GTX 20W50), antifreeze, brake fluid and transmission oil at Walmart. I also use Walmart store brand ATF in my forks. Can you tell that I'm a cheap bastard? My attitude is that by keeping the cost of maintenance down, there is less chance of deferring it. I go through everything on my bikes every Fall at the end of the season.
That "slime" in the cooling system may be the result of mixing different coolants. You might want to do a good drain and a couple of flushes with distilled water before refilling the system with a good long life coolant. I use Peak, but almost any brand name stuff will work.
Get a large bottle of DOT 4 and plan on running it all through the brakes system to completely flush out the old fluid. For brake pads, avoid organic, they are filthy and will keep the wheels dirty. Look for ceramic. Get them on eBay for the best price. Make sure you get the right year for your bike. I found out the hard way that not all years use the same size pad. Some years use pads that are a mm larger than other years. They look almost exactly the same, but the big ones can be a bear to install if your bike doesn't use them.
