I have a Tamron SP 70-200MM F2.8 and a Nikon AF-S 70-300MM 1:4.5-5.6G, I will use the Nikon vs the Tamron 90% of the time. I like the longer zoom and the lighter weight of the Nikon. You can get a refurbished nikon 70-300 and many other camera goodies with in your budget.
The 70-300mm does a nice job, but once he goes to shoot a game indoors, it's not going to help much