Can't shift front gear, despite loosening, tightening, etc. What could be wrong? Disclaimer: I'm clueless and am piecing together what little information I know from reading this site and from watching YouTube 'intro to repair' videos.
I've been gifted a used mountain bike, my first since I was a kid. It had a few issues, though. I managed to fix the brakes, but the shifter is giving me problems.
The bike in description is a Giant Boulder SE from the 1990s. It's got 3 speeds on the front and 7 on the rear, and operates using thumb switches.
Main problem
I can't switch gears on the front at all. By loosening the shifter cable a little, I gained the ability to click through the gears, but the derailleur/chain never makes any changes. It's always stuck on the outermost ring of the front... set of rings. (Cassette, I think.)
I think the larger ring represents a higher gear, but clicking through to top gear and then lowering it back down to 1 does nothing.
At top gear, the shifter cable tension is such that I can get a finger between it and the frame, and wobble it about 5mm, but it's still taught. On first and second gear, the cable is much looser.
This is a problem because riding in first gear kind of sucks, and by kind of, I mean really.
I have tried adjusting the limit screws. At first, I saw some movement, but now tightening or loosening either screw seems to do nothing at all.
Secondary problem
The rear shifter starts at 7th/top gear and 'releases' downward, towards gears with less resistance. I can go between 5th, 6th, and 7th gears, but trying to go any lower than 5 won't work, as there's just too much resistance from the shifter.
I don't think this is a huge deal, because 1st-in-front-5th-in-rear is a plenty low enough gear to get started/climb the kind of hills I'll be climbing anytime soon. But I thought I'd mention it anyway. My instinct is that this means the cable is too taut, but my instincts are garbage because as I said, I'm clueless.
Any help here would be really appreciated. Can't really afford to take it into a shop or that's what I'd do.
First note that if you get full movement in your rear gear shift you will probably have all the range you need. I think this would be EASIER than working on front derailleur. SO I recommend working on your Secondary Problem first! :-)

NOTE: Allow plenty of time for fiddling with your gears. Do it in a warm dry place when you have time so you can learn how things work in a relaxed manner. The less work you have to do the better! :-) Take it step-by-step. Adjusting loads of things in one go without having done it before could leave you with gears not working at all!  

Main Problem
Three sensible things and one that is not for the faint-hearted: 


CLEAN and OIL the front derailleur. Does the front derailleur itself move up and down (in and out) if you push and pull it with your hand (or you might need to use foot!)? It should move babck itself pulled by the spring. Clean the derailleur - it can get clogged with dirt - and oil it after cleaning. Because your cable moves but your derailleur doesn't this is the first thing to do.
CLEAN and OIL the cable Does gear cable move freely and pull on the derailleur properly? If you pull on the cable strongly with your (gloved) hand does the derailleur move up and down? Rust or dirt in cable might cause it to get stuck. Clear out cable with oil and take it off bike and lots of manual movement. Replace the cable if it is rusty or damaged.
Adjust the limit screws on front derailleur. The small screws H-stop and L-stop. One is top limit, the other is the bottom limit. They prevent the chain jumping beyond the rings and getting stuck - which is bad.
NOTE: Lowest gear is small cog on front, large cog on rear. 
NOTE: Highest gear is large cog on front, small cog on rear. 


turn your bike upside-down or put on stand or lift where you can freely pedal and also operate the gear lever


NOTE: You can do L-stop or H-stop in whichever order you prefer. It doesn't really matter.


move gear lever to lowest setting
pedal(gently) and adjust L screw until chain moves onto lowest gear (smallest cog)
pedal(gently) and check you can click up and down from lowest into next gear
move gear lever to highest setting
pedal(gently) and adjust H screw until chain moves onto highest gear (biggest cog)
pedal(gently) and check you can click up and down from highest into next gear


SECOND-LAST STEP: pedal(gently) and move to highest and lowest gear. You may have to adjust the stops again.

FINALLY: pedal(gently) and click all the way up and down through the gears.  If something is not right then you can repeat the process from start - you will soon get a better feeling for when the stop is adjusted correctly.


More info: http://sheldonbrown.com/derailer-adjustment.html http://www.instructables.com/id/How-To-Properly-Adjust-The-Front-Derailleur-On-You/step3/Front-Derailleur-Limit-Screw-Settings/


(Probably better NOT do this!) It is possible that the front derailleur attachment to frame is not right. Maybe it got knocked. You could unscrew and adjust height on frame or angle. BUT it is VERY tricky to get things working again if you move it! The attachment is usually very tight and not prone to being knocked into the wrong angle.


Note it is tricky to get the limit screws adjusted properly. But learn how to do it and it's very useful and will save you time with any bike you own.  

Secondary Problem Which I think is both EASIER to improve and will give you more BENEFIT.
Similar three things (1. 2. and 4.): 


CLEAN and OIL the rear derailleur. Springs, moving parts, screws and cogs. This will help loosen things up and gear cable will be able to pull better.

Check movement of rear derailleur. Move the derailleur in and out using your hand or foot. Does it line up with gear cogs correctly? The rear derailleur is prone to getting a bit bent or twisted. Some brute force can help (of course applied in the right place in a gentle mechanically sympathetic way!).
CLEAN and OIL the cable As in main problem. I think your rear gear cable is moving okay though.
Adjust the limit screws on rear derailleur. Small screws H and L again and another indexing adjustment screw. THIS is probably what will actually solve your problem!. 
NOTE: Lowest gear is small cog on front, large cog on rear. 
NOTE: Highest gear is large cog on front, small cog on rear. 


turn your bike upside-down or put on stand or lift where you can freely pedal and also operate the gear lever
move gear lever to lowest setting
pedal(gently) and adjust L screw until chain moves onto lowest gear (biggest cog)
pedal(gently) and check you can click up and down from lowest into next gear
move gear lever to highest setting
pedal(gently) and adjust H screw until chain moves onto highest gear (smallest cog)
pedal(gently) and check you can click up and down from highest into next gear


SECOND-LAST STEP: pedal(gently) and move to highest and lowest gear. You may have to adjust the stops again.
You can repeat the process from start - you will soon get a better feeling for when the stop is adjusted correctly.

FINALLY: pedal(gently) and click all the way up and down through the gears. If chain is not happy to stay on cog for some gears then adjust the indexing screw.  


More info: http://sheldonbrown.com/derailer-adjustment.html and http://www.parktool.com/blog/repair-help/rear-derailler-adjustments-derailleur