You will do fine with the Tamron (labeled as a Promaster) 70-300mm if you:
Use it where you have a lot of light like outdoor sports.
If hand held, use a shutter speed of at least 1/400 sec.
Always use F/8, and at or near 300mm F/11. Crank up the ISO if necessary.
I realize these are very very constraining settings, but if you follow the advice, you will be very pleased with the images.
You may not want to belive this, but this lens is likely better than the Canon EF 75-300mm F/4-5.6. Do not sell this lens to get the Canon EF 75-300mm F/4-5.6 (the Canon EF 70-300 F/4-5.6 IS is a different story, that is a very good lens). The Canon EF 75-300mm F/4-5.6 will not improve your situation (the Canon does focus just a bit better, but in good light the lens you have works fine). The Canon EF 75-300mm F/4-5.6 is marginal at any focal length at any aperture. At least the lens you have has some setting that will give sharp results.
In addition, it is likely better than the Sigma mentioned above at or near 300mm. The Sigma is a better lens at 200mm and below (you can use other apertures with the Sigma at or below 200mm and get nice results.).
Enjoy! Lon