Car rental is definitely an option. Local rentals can be quite cheap ($20 a day) but you'll probably pay a premium to drop it off in a different city.
I'm not sure it will be possible to do a rental that you drop off in a different country though. Its usually fine to cross the border between Canada and the US in a rental (but NOT into mexico if you're in the US), but I don't know if you can drop it off there. Still, most of the rental companys operate in both countries so you could contact them and ask. I'd just check what rental agencies exist at the airport in Toronto and then check the airport in Austin. Once you've got a common list start making calls.
If you can't do international, you could do 2 rentals and take a bus across the border. There's a bus operated by the company that owns the tunnel between Detroit and Windsor that should work out for this. You can drive from Toronto to Windsor, then rent another car in Detroit for the rest of your trip. Opening day for baseball should be in April, you might be able to catch a game in Detroit while you're there.
One thing to keep in mind on car rentals is that you generally need to be 21 to rent a car in the US (I don't know about canada). And if you're under 25 there will be a daily surcharge, usually $20 a day. That could definitely be a problem.
Just as an experiment I tried a rental from Hertz from Detroit to Austin, it was $1222 per month. At that price you might consider just buying an inexpensive car, depending on how long you plan to go for.
I've also read (last time I was there) that some cities do this thing where you can just pick up a car from a certain point and help "move it along" to its destination or something...
I've heard of this as well, and I used to know a couple of sites but I can't find them anymore. I remember there not being a whole lot of activity. You might be able to work something out on craigslist but it would be unreliable.