more important is what sports are you shooting. i shoot sports semi-professionally, meaning i make a lot of money doing it but its not my only income source. depending on what sport you are shooting calls for what focal length and more importantly what lens you would need to do a decent job.
while 70-300 is a great focal range for sports that lens is not intended for sports use and therefore combined with your camera which i have not tons of experience shooting with might result in slow autofocus since the lens has a good amount of glass to move around.
55-250 will be a money saver and offer similar quality and easier to use.
also VR is worthless when shooting sports because your shutter speeds will be soo fast you dont have to worry about camera shake.
any of the cheap super zooms will work nicely but honestly at a certain point if you care about quality the camera body will be the factor that holds you back the money so honestly but the cheapest super zoom you can get for the money because quality wise they will all provide the same results.
in my opinion the 55-250 will be fine and just crop your images if you need to. unless you are shooting sports from the stands in which case i suggest just not shooting since that is a terrible angle for sports photography. unless you are on the field or court dont expect too many awesome shots. just being realistic.
here are some of the shots i have taken in the past. idk how much longer my free portfolio will be up but you might get some inspiration or something. http://brian_r.500px.com