There are two current and several previous models of 75-300.
For sports photography, USM lenses would be best... STM would be pretty good... micro motor lenses are too slow for fast action or tracking subjects.
The
EF 75-300mm III USM is okay. Just "okay". Not the greatest image quality, but decent focus performance, low cost. No image stabilization. Cheap build. Slightly better "feel" than the non-USM (below). It sells for $188 refurbished at the Canon online store.
The
EF 75-300mm III (not USM, nor IS) is the cheapest ($160 refurb) and weakest telezoom that Canon makes. Image quality, build, focus performance are all sub-par. No image stabilization, either.
FYI, here is a not very complimentary review of the better of the two... the EF 75-300 III USM:
http://www.the-digital-picture.com/Reviews/Canon-EF-75-300mm-f-4-5.6-III-USM-Lens-Review.aspx There also was an EF 75-300mm USM IS... but I think it's been discontinued. It got stabilization, but if anything it's image quality is even worse (stabilization requires additional elements, so no doubt the optical formula had to change to accommodate it).
The best thing about the Canon EF 75-300mm in all their versions is that they're cheap. Unfortunately they prove the old adage, "You get what you pay for."
For not a lot more money ($200 refurb), the
EF-S 55-250mm IS II is a sharper lens with less chromatic aberration, and - especially - with stabilization. Not the greatest build quality (plastic), and not the greatest focus drive system, but it will make better images than either of the 75-300mm lenses. The IS alone will help a lot with a telephoto zoom, especially at the 250mm setting and on a crop sensor camera.
The
EF-S 55-250mm IS STM is a nice step up for about $50 more ($240 refurb), with all the same benefits as the non-STM model, plus better STM (stepper motor) focus drive performance (smoother, quieter).
Here is a review for the EF-S 55-250mm IS STM:
http://www.the-digital-picture.com/Reviews/Canon-EF-S-55-250mm-f-4-5.6-IS-STM-Lens.aspxThe Digital Picture doesn't have a review of the EF-S 55-250mm IS II... But Ken Rockwell does:
http://www.kenrockwell.com/canon/lenses/55-250mm-is-ii.htmThe next step up are $600+ 70-300mm lenses, three version (IS/USM, DO/IS/USM, L/IS/USM). Then beyond that there are the four premium 70-200mm (all L's... two f4, two f2.8, both with and without IS), starting at around $700 and going up to and over $2000.
Presumably you are looking for a low cost, lighter weight lens. If it were me, for my money I'd go with one or the other of the EF-S 55-250mm... They are still among the least expensive available. But spend just a little more and get much better images as a result.
I can't compare with the Tamron lens you mention. Check on some of those websites and elsewhere, you might find a review of it. I have used some Tamron lenses and one thing I'd almost guarantee if it's a lower cost model... it will not focus as quickly and surely as either Canon STM or USM lenses will, making it less fun to use for sports photography. Tamron does make a few USD lenses, that give focus performance similar to Canon USM, but they typically are the more expensive Tamron models.
With any of these lenses, using a T3i for sports your daughter will probably want to set it to use only the center AF point. It's the only "cross type" point in that camera model... which makes it faster and more responsive than the other eight AF points. Set the camera to AI Servo, and she's ready to shoot cross country. Indoor hockey is probably going to be challenging, as most arenas are not terribly well lit. This is where expensive f2.8 lenses are popular (IS helps steady shots by counteracting camera shake at slower shutter speeds... but it can't do anything about subject movement that might cause blur at those same shutter speeds). Still, depending upon conditions, she can probably boost the ISO in her camera to 3200 and get some decent shots.
EDIT: I just did a quick search and find no reference to Tamron 75-200mm anywhere.
There is a Tamron SP 70-200mm f2.8 lens. Again, I've never used one with my Canon cameras, but it's a premium lens that might blow away any of the above in terms of image quality, build and overall performance. If your daughter has one of the Tamron 70-200/2.8, she probably wouldn't be very excited about any of the above lenses. In particular, the current Tamron SP 70-200mm f2.8 USD VC is very highly regarded (Canon's $2000 EF 70-200/2.8 IS USM Mark II is better... but a lot more expensive).