Lots of people have already given great advice. I don't see where you posted what type of car you have though. Unless you have a pick up truck or real SUV, the bike carrier is probably going to be too much tongue weight on the tow hitch.
If you have towing ability, I would strongly recommend renting the U-haul motorcycle trailer. It is $15 a day, and it will give you a chance to evaluate your options before spending money on buying something pricey.
In terms of initial trackdays, I think you can keep it fairly low budget. Run some Q3+ or Q4 tires, and don't stress. Besides from trackday fees and tire costs, the biggest costs that people often don't think about are:
lodging
gas
food
If you're living the comfortable life, you'll easily spend $300 just on these three items per weekend.
Depending on how far you are towing, the gas money for your car and the bike does add up. It is really an afterthought, but I do spend probably $80+ on gas alone per weekend.
Lodging is another huge cost, as others have mentioned. I sleep in the back of a station wagon now. When I did not have a tow vehicle, I used to rent Ford Transit vans from Enterprise commercial trucks and just string up a hammock inside. Otherwise, you're likely spending $50+ a night for a room. For a weekend, you will easily spend $100 on a room.
Food. Either you can pack your lunches and drinks in a cooler, or you can buy stuff nearby. Restaurants for 3 meals a day. Drinks after the day is over. Snacks. Water. If you eat out and buy everything there, you'll easily spend $100+ for a weekend.