lens choice help

[ronin2307]

Well it didnt take long to realize that my Canon kit lens will just not be quite useful in ALL situations, hence my desire to buy a telephoto zoom lens. Since I am extremely new (yet borderline addicted) to photography, i thought i'd ask for some advice.
I realize that most people invest in lenses vs bodies since those are useful much longer. With that said i am debating if i should opt for the

Canon EF 70-300mm f/4-5.6 IS USM
or the
Canon EF-S 55-250mm f/4-5.6 IS

I know the diff between EF-S and EF and also the focal range but I am wondering if the price diff is worth it for the first one. I have read reviews of both lenses and they hardly seem to suggest anything bad about them (at least for a hobby photographer like myself) but would like to hear opinions here.
My subjects are typically my little girl and whatever trips my trigger finger on hiking trips, sport events for little nephews etc
any opinion/suggestion will be greatly appreciated.
thx

[CaraRose]

My understanding is the 70-300 is a better quality lens than the 55-250.

I have the 55-250 and it is a nice little lens, although after using L quality glass I now notice the lower image quality

[ronin2307]

slrgear seems to think highly of the 70 lens, especially when compared to L glass. seems like this may be a better lens to grow into.
some people say that it is faster, which i am assuming is in reference to AF since the f-stops are the same (unless i have completely misunderstood that concept)