Any level of components will shift fine without autoshifting if it is adjusted correctly. Even no name Wal-Mart components shift OK. This is all assuming that something is not actually broken or worn out.
The only reason that upgrading will fix it is that you may replace a faulty (money well spent) or misadjusted (waste of money) component.
Is everything in good working order? Including the cable and housing being clean and lubricated, with no kinks, etc. Shifter - Is that working OK? Dropout straight and not cracked? If the dropout is replacable, make sure it is tight. God forbid, but make sure you don't have a crack in the chain stay. This can let things move around and make the bike shift.
Here is a rough check to see if it is adjusted correctly. Put the bike in whatever gear you are having the most problem with. Get behind the bike and look at how the top pulley and the cog line up. They should be pretty much perfectly in line. If not, you are simply out of adjustment.
If you are mechanically inclined........
I reccommend that you go to the parktool websight and look at the instructions for adjusting a rear derailleur. Read them completely. Then, I would start from scratch by releasing the cable, and setting the barrel adjuster to it's middle position. Now you should be at "neutral". Set your high and low limits first. Then set the cable according to the instructions.
It is not hard at all. Just make small measured adjustments. Such as one turn on the barrel adjuster. If it gets worse, put it back were it was and try the other direction. Only adjust one thing at a time.