Like Xavier said, your rim is shot. You will notice that the "drive" side or cog side spokes are under higher tension than the non-drive side. This is necessary for something known as wheel "dish". Spokes on the drive side almost always pull through first compared with the non-dirve side.
This is not the fault of your bike shop. Spokes are under high tesion and riding causes a wheel to "destruct" (fatigue/fail) every time it makes a revolution. All bike parts will fail eventually. Regular inspections are a part of good riding.
I have used Mavic rims for many years. They are the finest rims available. Some bikes come with cheap rims that fail rather quickly. Get a set of quality wheels built.
For training, I always use 36 spoke Mavic Open Pros with DT spokes. Higher spoke counts will make for more weight and wind drag (use heavy tubes/tires also), but it will make you stronger as a rider. For an important ride, throw on a set of quality, lighweight/aero wheels. You will be faster!!!
You can ride on the wheels without problems most likely. I have had HUGE chunks pull out of rims many miles from home. If the wheel goes out of true so bad that the tire rubs against the frame, you will have to do an emergency roadside truing session to get it straight. You don't have to get it perfect or anywhere near it, just get it off the frame. You might also have to loosen up your brake cable for the tire to clear the caliper as well. Spoke wrenches come in different colors for different sized nipples. Have your shop hook you up with the correct color/size tool if you are unsure.
Also know the basics of wheel truing before riding on a bad wheel. Tighten the side of your desired direction. If the wheel rubs the frame on the left, you need to go to the right. So, tighten a spoke on the right side in the same area that the wheel is rubbing on the frame. Sometimes a nipple will be frozen where you need to turn it. Move to the next nearest spoke and try again. You can also loosen a spoke to achieve a desired effect. Loosening a left side spoke will move the rim to the right.
Sometimes, your wheel might be so warped from a crash or whatever, that it cannot be trued with a spoke wrench. This requires the "brute force" method. Take the wheel out of the bike. Put it against something firm like a concrete curb and carefully thust your foot agaist the middle of the spokes until the wheel bends back into something of a resonable shape. Don't throw your foot into the spokes like a kickboxer. With mild tension agaist the spokes to start, gently but firmly increase the pressure until the wheel get back into line.
Throw these wheels away once home.
I've seen Jobst Brandt post on this board. He wrote a book about bicycle wheels and building wheels. Great reading if you are into wheels!!!
Ultimately, it always comes down to how far you want to push your bike. If you are not a good mechanic, stick with short rides until you get some new wheels. But then again, how will you ever become a good mechanic unless you go through these challenges??? Taking your bike to the shop every week gets old real fast!!!
Know your limits, but don't be afraid to push your boundries and pick up new skills!!!
Good luck!!!
