ON my new bike I noticed the loud clunking I've been experiencing seems to be a slipping chain that keeps falling off the largest rear cog when shifting. I've experimented with pushing in the derailer a tad and it seems to allow the chain to stay on. Supposedly when I left the shop after buying this they set it up to make sure all was ok but apparently they didn't do such a hot job as the shfiting has been difficult and clunky. what do i need to look at to ensure total shifting capability on the rear shifting. BTW I assume that between from and back gears I can run largest to largest ..yes? This should offer the easiest peddling?

On the front-to-back: smallest chain ring (front) to largest cog (rear) will offer the easiest pedaling, but not much speed. As you go smaller rear, or larger front, pedaling gets harder, as speed goes up.

As far as shifting, if you have ridden more than 20-30 miles on a NEW bike, you may have gotten some cable stretch, which is normal for new cables, but will make shifting to the largest cog more 'iffy'. Your shop should do a free 'tuneup' as part of the purchase price--a good reason to buy from a LBS. If you bought it from someplace besides a Local Bike Shop, you are on your own for tuning it up. Not too hard for the rear--turn the cable adjuster at the Rear Derailleur about 1/4 turn counter-clockwise, and test ride/shift it. If not enough, try another 1/4 turn.

Running largest to largest will produce an intermediate gear. For the lowest gear you want smallest pulling the largest. The basic math for gear ratios is driving sprocket divided by driven sprocket, so either a larger chainring or a smaller rear sprocket will have a similar effect.

While it's absolutely critical that the chain can loop the two largest sprockets with an inch to spare (so as not to destroy the drive if you accidentally shift into it) it's not generally a good idea to use the combination much. The crossing angle inner to outer (or the opposite small/small combination) causes increased friction and wear, and a more efficient comparable ratio probably exists with a more efficient combination.

If you have a few minutes to spare you can count the teeth in each sprocket front and rear, and do the math for the ratio of each combination to determine the best combinations that will give you a good selection of gears without needing use the inefficient crossed pairs.

well after much fudging around I got the chain back on and technically the derailer is lined up but still a problem exists. The derailer with or without cable attached pulls to the right where the midlle spring arm mechanism is. If I hold it rigid and pedal all goes smoothly but as soon as I release my grip it pulls off the sprockets and to the right. Is there some other adjustment I'm missing that stabilizes the derailer arm?

The way the RD works is the cable pulls it in toward low gears against a spring that returns it to the right when cable tension is released. It's possible it's a simple centering adjustment, though that should affect all gears. Or the device in your levers that engages the clicks is weak and the spring tension is enough to pull it out of gear.

Given that it's a new bike, you should return it to the point of purchase and have them resolve the various issues, plus better explain to you basic utility, and user adjustments.