Toyota Repair: 99 corolla manual transmision, corolla ve, rubber tip

Question
i was just given a 99 corolla ve as a gift it has about 79000 miles. i noticed that after i shift into gear wether 1-5, if i accelerate, the engines sounds like its being reved and not necesseraly speeding up, almost as if if neutral. after a couple of seconds it seems like the trans kicks into place and then speeds up, but this is a constant issue in all gears. I also have a question about the radiator, it had a small leak so i bought one of those stop leak products with the pellets in it a poured half of it in as indicated on the bottle. The leak seams to have stopped, but when i was able to get under the car and look at the radiator i noticed what looked like a small valve with a rubber end cap on it. the rubber end cap is broken and this might be where the leak was comming from but maybe after i poured the stop leak it might have temporarily sealed the opening to the valve. Could this broken rubber tip at the "valve" have been the cause of the leak, and if so where could i replace this piece and what is it called. Thanks in advanced for the help.

Answer
The problem is that the clutch is slipping, it needs to be replaced. I recommend that you have a toyota dealer do this repair, it will probably cost around $600-700 but since you got the car for free it's still a good deal.
Not sure what the problem with the radiator is but I would have it checked while it's in the shop.