There is no good reason for you to be running cross-chained. That causes increased wear on the chain/sprockets, increases pedaling resistance and tends to exacerbate any anomalies present in the system.

You can achieve the same effective gearing by running on the middle chainring and one of the "middle" cogs on the back.

Try running on the middle chainring and the #4 cog in the back. That is how I start out on my 24 speed MTB. That will allow you 2 more "higher" gears on the cassette as you gain momentum. If you need more speed than what you can get from the 5th or 6th cog, then you could shift to the large (outside) chainring.

For more climbing power on the middle chainring you would shift down to the 3rd cassette cog. If you need even more climbing power, then it would be time to shift to the small (inside) chainring. That will give you 3 climbing gears on the smallest chainring. I would not normally want to run higher than the 4th cog on the small chainring.

When your gears are set in the middle chainring - 4th cog gearing, check your front derailleur. Your chain should be centered or slightly to the outside in the derailleur. If it isn't, use the cable adjuster to achieve centering. You can use the adjuster on the rear derailleur cable to make sure the derailleur is aligned with the proper cog sprocket.

If you still have problems with the chain slipping off the chainring after all that, adjust the end points of the front derailleur as described by corrado33 in post #6. There are similar adjustment for both the low (L=inside) and high (H=outside) end on both derailleurs.