You also need to avoid shifting the front chainring at the last minute when you are exerting alot of pressure on the pedals (it is always best to let off on the pedal pressure on the pedals when shifting); as cny-bikeman said
Your bike has two or three ranges of gears, one for each chainwheel, but they overlap, so the extreme combinations are not really necessary. Another combination can be found that will feel about the same (and get you going the same speed).
Since your riding MTBs you probably have 3 front chainrings; look at these as riding modes. The small chainring is for climbing hills, the middle chainring is for around town and the large chainring is for going fast on the flats or downhill. The nature of the forces on chain chain allow you to change the rear gears (cogs) even without letting off pedal pressure (unless the pressure is extreme), without unnecessary wear or damage. However shifting the front chainrings under pedal pressure can wreck havoc on your crankset; it can delay or resist the shift, bend chainrings and/or break the chainring teeth, cause what is called chain bridging, where the chain is locked mid shift, attached to both chainrings, and/or brake your chain (my guess is this the front shifting is where your bike problems lie). In the future you should anticipate when you will need to shift the front chainrings. If you are riding along in the second gear and there is a hill coming up, shift into the small ring and then shift into some higher gears in the back, so when you get to the hill, you will already be in the right front chainring (or riding mode) and will be able to down shift the rear into an easier gear. The same for shifting from a small chainring to the middle (or middle to large). In stead of up-shfitng the rear cogs, until you run out of gears and have to shift from an extreme chainline while you are spinning madly, shift into the next biggest chainring right when you need a higher gear so you be in the right chainring (or riding mode) for a flatter ride.