I started with the 75-300 and it's a reasonable lens to get started with. It's small and light. Remember to use a beanbag or tripod to get sharp pictures.
I'd also invest in a daylight or UV filter to protect the front element.
Let's say you could wait 6 months (I know for a 13 year old that's a long time) ....how much more money would you be able to save or get together if it's possible?
Bearing in mind that was there a year ago, I think it might've been sold?
But both are good, the sigma's a good alternative but that Canon you suggested from Amazon's a better bet IMO! The canon is £1000 new, so I wouldn't hope one second hand to be too much less!
if its for birds I would go for the 400 f5.6 prime canon lens,best lens around for the canon,but costs £1000+ problem with sigma lenses are ,you never know if you get a good one or a bad one , I had 2 copies of the stableised 150-500,one wouldnt focus at all and the other was erattic with getting a sharp shot and had to pay to have it calibrated.
If you've got the budget i'd get an Image Stabilised lens, I started on the Tamron 75-300mm which isn't IS and then I got the sigma 120-400mm which is. I've been in exactly the same position 2 years ago, but I think if I started again I wouldn't buy my old 300mm, i'd wait and get an IS lens. The sigma range are expensive but trust me its worth waiting, and i'd look second hand on ebay or just on websites like this one, i'd say its well worth biding your time, not necessarily buying new and looking for bargains! worth bearing in mind that whilst Canon lenses are better quality they're not that much better than sigma!
It would be a good start to get used to longer lenses. It's quite light too. I know it may be a bit more expensive but you should also look at the Canon 55-250 4-5.6 IS. This has image stabilisation and has always had very good reviews, it might be 50mm shorter but the image stabilisation will help a lot. I started off with a sigma 70-300 lens which I still have in my collection.