I just shortened chain on my new bike again. Turns out I put it on way too loose. I think particularly with a compact double (because there's a big difference in ring sizes from 50-34 vs. 53-39), you really need the chain to be absolutely as short as possible. That means big ring to big ring plus one link. You really do not need any slack for the derailleur at all - the derailleur can move to the chain. And you will never really ride big-to-big. The problem I was having was chain slip when I was near the little-to-little rings. I now know this was because I had too much slack in the chain (riding little-to-little on a compact double with a pretty big cassette range isn't a great idea in any case).
Some tips on buying a good old used bike : I think the best bargains are actually in pretty recent low end bikes; 2-5 years old, generally aluminum, they just have zero sex appeal so there's no markup. You can get used road bikes from chinese brands like Giant or the Performance Bike generic brands with Tiagra level components for $200 and they will work better and be lighter than very expensive old bikes. If you want an older sexy steel bike, here are a few tips :
1. In general you want to look at bikes from about 1985 - 1995 ; after 1990 would be preferred. If you get a bike that's Shimano components post about 1985 , that will give you compatibility with the whole modern generation of components - all the recent generations of 8 & 9 speed pretty much work back to that era (new 10-speed stuff doesn't). Avoid old Dura Ace stuff, though you probably won't see it. Avoid the older "ten speed" era bikes. Avoid French bikes at all cost because they have weird and annoying compatibility problems that make parts much harder to find and more expensive. Avoid old Campagnolo unless everything on it works perfectly.
2. I think Japanese bikes like Bridgestone and Miyata are a good way to get quality stuff without paying the "sex appeal" markup of Italian bikes. Brand names and collector's items will have big markups that you don't need to pay, you can find great stuff that's just as cool if you do a bit of research.
3. Make sure the frame is a more modern higher quality type of steel that doesn't weigh a million pounds. Obviously if you can get a real sexy frame that's like Reynolds 531 or 853 or one of the better Columbus or Tange steels, that's nice. Avoid the pre-1985 straight steel super heavy shit that's in bikes like Schwinns, Pinnacle, Peugot, etc. Look for "butted" on the stickers. Don't get sold on paying a premium just because of a "Columbus" (or whatever) sticker though, those are just steel tubing brand names and they made a whole line of steels, some cheap, some premium. But really don't focus on this other than avoiding the super old thick crazy heavy shit. If the frame is good, make sure it fits you nicely, and then you can always replace components.
4. Look for rust, dents, big gouges. You don't want to deal with any of these. A tiny bit of surface rust on one or two spots of the frame is not a big deal, usually you can just scrape it off.
5. Check for parts being frozen. Take your tools with you. In particular check the stem is not frozen in the steerer and the seat post is not frozen in the seat tube. Take off the pedals and make sure they are not frozen. Again it's not worth buying a bike that has these problems. Often people who owned these old bikes never greased them or serviced them a single time in the last 20 years.
6. Bikes that are badly adjusted can be a nice bargain. They will seem to brake terribly and shift badly, etc. The owner thinks it's "in bad shape" but you can tell it's structurally fine. Usually when I buy an old bike I plan to throw away the tires and tubes, brake pads, and cables. Replace them all it, clean it, lube it, tune it, and it works like a dream. That overhaul will cost you about $100 in parts.
7. Bikes sold with "extras" can be a big bargain. You'll see naked bikes for $150, and then you'll see bikes with fenders, racks, bags, lights, etc. for $250 ; those parts alone are worth $250 if they're in good shape. Sometimes just one component on a bike can be worth the whole value when people don't know what they have; something like a premium saddle is worth $100+ and you can find them on $200 bikes.
8. I recommend looking for a frame that's versatile. Look for braze on fender mount eyelets at the fork and rear dropouts. Old touring frames are great to turn into commuters because they have the good rack-mount brazeons on the seat stays. Look for brakes with clearance and space for fatter tires and fenders. You might never use any of these things, but it's good to have the flexibility so that you can do whatever you want.
Don't try to save money on inner tubes. The difference between a cheap tube and a good won is actually quite large. A cheap tube is around 125 grams, a good one around 75 grams. 50 grams may not seem like a lot, but shaving 100 grams = a quarter pound off a bike would cost your $50 - $100 , so saving $5 on tubes is highly irrational.
Tires are even more important than tubes. The same is true of cars of course, I've written before about how retarded it is when people buy a $50k sports car and then put cheap tires on it. It's just a radically bad price-performance decision. Of course those people don't actually care about performance, they only claim to; in reality they are spending for image, so maybe it is a logical price/image decision.