Whether you upgrade components or replace the bike outright depends upon how many miles you are doing and how hard those miles are on your bike.
Typically, if you need to replace one part of a particular component group (e.g. Tourney) due to its lack of durability or performance for the type of riding you are doing with a higher-end part (e.g. Deore LX), then you will probably be replacing other parts on the bike pretty soon as well.
If you are wearing out or breaking parts on a somewhat regular basis, then you may want to take a look at what bikes others are riding who ride in the same places you ride. Then head to your local bike shop for some test rides (and your bank for some budget calculations).
You have to be careful upgrading parts on a sub ~US$350 bike (typically a hi-tensil steel or 3-tube Cro-moly frame and comfort suspension fork) since you can easily spend enough (or close to it) to buy a brand new bike outright with the same components AND a much nicer frame and fork (e.g. full cro-moly frame and entry level performance suspension fork).
Typically components in the Deore or Deore LX groups provide a very good price to performance ratio for most mountain biking in a variety of conditions.
Just try to keep in mind when making upgrade versus buy decisions that how the bike performs depends on how the frame, fork and components work together as a whole. So just be conscious of significant mismatches between frame, fork and component performance and durability.
Also note that the frame, fork and components that make up a bike are much less expensive when purchased together (i.e. as a bike in a bike shop) than trying to purchase each bit individually.
Good luck!