My rear derailleur apparently isn't aware of the my lower gears
 

I played around with my rear derailleur trying to learn. I looked at various YouTube videos, looked up Park Tool and other pages, learnt about H / L screws, the cable itself etc etc.

I tightened the cable keeping the chain on the biggest wall in the chain ring and the smallest in the rear sprocket, tightened the H screw correctly and tried shifting upwards. It shifts correctly for 2 gears, but it simply refuses to go to the top 3. (3 chain rings and 6 sprockets) .

I can see the derailleiur move a little but it is not enough to pull it upwards. However, it switches perfectly between the smaller 3 rings (Note: This happens on all 3 of the chain rings)

What am I doing wrong? (Note: I could take this to a bike shop and have it repaired, but this is a old bike and I am just trying to learn the mechanics here)

Additionally, what's a good understanding-bike-mechanics book you guys would suggest? I have seen plenty of YouTube videos and read online, and the one small "bike maintenance" class I have in my city teaches how to change tubes after a puncture and some minor lubing and stuff ..

Did you have the shifter in the smallest cog (highest gear) position when you tightened the cable? If not, unclamp the cable, place the chain on the smallest cog and shift the shifter to the high gear position. Then reattach the cable. If that doesn't do it, your low limit screw is probably way too tight.

+1 It's one of those issues.

http://www.sheldonbrown.com/ also has lots of good articles to learn from.

Add to the above - make sure your lever is in the high gear position when attaching the cable. Check before disconnecting the cable by drawing the bare wire away from the frame like a bow string as you click the lever all the way to high. Then disconnect, thread the trim adjuster all the way in, pull down all the slack and start again.

Attach the cable in high gear, click the lever one position and adjust the trim until it makes the shift and center it. If the low (inner limit) is set correctly, all the other shifts should be fine. Adjust the inner limit, not using the shift lever, but using the bowstring method to shift to low, setting the limit so it allows the last shift with no possibility of the chain going beyond. When all is close adjust the trim for good shifting in both directions.

Thanks a lot for the advise! I did as you suggested ... and it seems to be working fine now !

Department of pet peeves.

Advise, verb, to give counsel, offer an opinion, make recommendations or give advice.

Advice, noun, recommendation or counsel, what is given when advising.

If I was thanking you guys for the information that was given to me (advice) as opposed to the act of you guys informing me (advise) wouldn't that make my sentence correct?

It would, but you used the verb form, advise, which prompted my response. If you wanted to thank us for the action vs. the act, proper syntax, would have been thanks for advising.

You probably know it, and this was probably a typo, but I see this mistake so often on the forum that I mention it every once in a while.