I have a soft sided tool bag made by Custom Leather Craft (CLC)
http://www.goclc.com/products/Work-Gear ... -Backpacks I found mine at KMS tools. Get one that keeps everything secure and zipped up. Obviously avoid the electrician style open top ones
I bought a cheap $40 four wheeled, two shelf cart to throw my bag on while in the hangar.
As an apprentice, showing up with a decently stocked tool bag is more than adequate. Every job will be different and until you know what the more experienced guys are using, you will likely waste money getting a big tool box or tools that you may never use. I know when I was an apprentice, the licensed guys were happy to lend me the more odd-ball or expensive/large tools the first few times one was needed. If it is a tool that you will be commonly using, you can then go out and buy your own.
As for buying tools, you will hear all sorts of advice. I personally bought Snap-on pliers, screw drivers, picks, and ratchets, Craftsman USA (buy from sears in the USA) wrenches, sockets and extensions, and everything else from places like Canadian Tire, KMS Tools, and even Princess Auto. Second hand is also a great option. Buying off an older or retiring AME or keeping an eye on craigslist and kijiji has saved me a bunch. I got a nice Fluke multi-meter for $40 of kijiji!
Weight is definitely a very important factor when getting tool storage. Besides the commercial airline weight limits, company aircraft will frequently be bursting at the seams with ops gear, oil, tools and parts when sent out in the field.
For airline travel, the best thing I have found is to take out things like hammers and large wenches and move them to your clothing suitcase, then putting the tool bag in a Rubbermaid action packer tote with light gear like coveralls and jackets for airline travel. If you travel with three bags, when paying the third bag fee, you are allowed to have that third bag be up to 70lbs with no additional overweight charge (Air Canada). Also, a personal baggage scale is a great investment!