Kits are nice but a large proto board and an oscilloscope with
a book like 10,000 Electronic circuits for experimenters was
better for me. Every basic circuit from simple one transistor
amplifier to complex logic and discrete components is there
to experiment with, and you can substitute your own devices
and components in seconds for real design and R&D.Plus if
you use scope you get a depth of knowledge that you just
can't get pre packaged. The Xermac Mosfet power supply was
designed by myself as was the Custom EDM servo drive and
about 100 others.(To long to list)

A great place to start is http://www.apogeekits.com, they have a wide arrangement of kits from beginning to expert. The site looks like it's from 1990, but they do a pretty good job of making it easy to find what you need. It seems like you have a little experience so I think starting on intermediate kits would be a good choice for you. Another good site to check out is http://www.electronickits.com/index.htm, they have some pretty cool things there too.

http://www.apogeekits.com/images/electronic_kits.jpg

http://www.electronickits.com/kit/complete/audi/ck1207.jpg

Kits are amongst the best ways to understand and learn about electronics. It is challenging and you put in practical terms what you want to do and learn. One site that offers several electronic kits that range from beginners to advanced is CanaKit. They have been in the market for over 20 years and has a good range of kits from FM receivers to stereo amplifiers. Check them out: http://www.canakit.com/

For computer related kits, there is TigerDirect, which offers an array of different kits that will suit any ones needs and interest. Check them out: http://www.tigerdirect.com/applications/Category/category_tlc.asp?CatId=31&name=Barebones

This answer may seem counter-intuitive to the rest, but I think your best "kit" might simply be to get your hands on a few older computers that you can take apart and look at. Computers are everywhere--they are on the curb on trash day, they are offered up at Goodwill truck trailers in our community, destined to be recycled, and they're very very cheap at garage sales and thrift stores.

When I was a child, my mother figured out pretty early that one thing that kept us entertained on a quiet rainy day was to take things apart. She would go to the Salvation Army and buy old wind-up clocks and when the time was appropriate, she'd set us at the dining room table and give each kid a clock, a cake pan, pliers and a couple of screw drivers. This isn't to say we necessarily learned to fix or make clocks, but taking things apart and looking at them is very educational.

We had other opportunities to dismantle things over the years, and I discovered a knack for fixing things in the process. As a 13-year-old, I had a summer job at a museum and ended up fixing all of their old antique cash registers. :) These are mechanical, not electronic, but the idea of getting your hands on things to see how they work carries over. I tend to be the office go-to person to fix devices for co-workers (printers, computers, scanners, and the photo copier in particular) when they're not working, and often times we don't need to call the tech.

So find some old computers, open the cases, and look around. See how things fit, do some research (you can look at the specs for just about any regular brand computer by entering the model number at the manufacturer site. They often times even have schematic drawings available. This is certainly the case for HP and Dell.) Be sure to learn what the usual steps are for working on electronics (work with it unplugged, push the "on" button before doing anything to discharge it, and be careful when working during dry/static electricity conditions. Don't touch the little shiny brass parts of any of the cards, etc.)

You might even want to try taking them apart then putting them back together again and taking them for a spin. If you start with a CPU that you know works, and tinker with it, then try running it again once it is put back together, you'll learn something.

This isn't programming--that's a different aspect of the process, though if you get an old enough computer or decide to put a very basic operating system in and practice with DOS, etc., you'll learn some useful background information that still applies to more current machines. Any time you call a techie (who may be 1/2 way around the world) and they want to do some diagnostic, have you ever noticed that they'll tell you to do a combination of keystrokes that bring up the black command window, and you'll be doing DOS before you know it.

Depending on your age and interest in electronic toys, there are lots of kits (some mentioned above) and there are also components that let you create your own devices. I'm thinking in particular of the Legos electronic components for building things that move, roll, lift, etc. And others are following this line of thinking. The components are part of a system called Lego Mindstorms. http://mindstorms.lego.com/en-us/default.aspx Take a look around the site and you'll see that they offer the bits and pieces to build all sorts of things.

My son discovered these in middle school and we ended up ordering lots of bits for various projects of his. He was lucky in that the local community college offered classes every summer for talented and gifted kids, and one of the classes was to do with Legos of this sort. I don't know how old you are (this program was for kids from 1st through 8th grades - http://www.tccd.edu/x1288.xml). Here is a site with all sorts of entries based upon Lego Mindstorms http://www.marioferrari.org/lego_mindstorm.html

This same son has had a lot of chances to get his hands on the insides of his computer, and for his high school graduation he wants to build his own computer instead of simply buying one like we did for his older sister. We're all looking forward to this project!

Good luck with your studies--if this is something that really interests you, then you're halfway there. Let your curiosity lead the way.

Finally, down here at the bottom, I'll see if this video loads properly: It has to do with building electronics from component parts, a design created by a man who figured it out while playing with Legos with his son.

http://www.livescience.com/common/media/video/player.php?videoRef=ns0063-bug_labs

The flash video won't load here, but follow that link for an interesting video. And if you Google "Lego Electronics" you'll find tons of stuff.

There have been some good answers here and I'd like to add to the list. Heathkit is still around, and has been making electronic kits for decades:

www.heathkit.com

Also, you should check out the 'zine Make for project ideas.

http://makezine.com/

You can also join a "hacker" club, or a group of people who like to build electronic projects, or take other things apart and learn about them. Some popular ones here in the northeastern US are the NYC Resistors and Fubar Labs in NJ.

http://www.nycresistor.com/
http://fubarlabs.com/

An excellent hobby that will further expose you to more electronics people and building, is amateur radio, also known as ham radio. Ham radio is for radio and electronics enthusiasts, and uses radio for communication rather than systems that require some sort of infrastructure, like telephone or the internet. Here is a good guide for getting started:

http://blog.makezine.com/archive/2009/08/theres_plenty_of_fun_to.html