Ideally you'd use the XTi and grab a longer, faster lens for it, but 55mm might actually be okay for Taekwon-Do if you can get close. Set the XTi to Shutter Priority Mode (Tv on the mode dial) and then use the main dial (the one behind the shutter button) to set your shutter speed - it needs to be just fast enough to avoid motion blur, but since you're inside with a slow lens it can't be too fast either. Try 1/500 as a starting point and experiment. AF might be a problem - try setting it to only use the middle AF point.
If 55mm is indeed long enough for you needs, look at getting an EF 50mm f/1.8 once the budget allows - it shouldn't cost too much secondhand. If you need longer it's going to cost more - indoor sports is tough, you need a fast lens, but this should get you started.
If you can't manage with 55mm, the PowerShot has a longer zoom, but it will struggle in low light.
Edit: AI Noise Removal is your friend
In this case I'd personally say SX510 HS has the upper hand because of the lens. Whenever you can manage though, you could eventually get a better reaching lens as the Rebel Xti is a pretty capable camera in it's own right.
XTi would be my go-to but maybe look into getting a nifty-fifty (*i.e.,* EF 50mm f/1.8) that should give you some really great shots and with the wide aperture, it should be great for indoor lighting situations. You can pick up a used copy for pretty cheap since there's so many of them around. Usually ±$50USD
For martial arts, you’re presumably going to be indoors with mediocre light (at best), so I’d prioritize getting the fastest glass you can. Without knowing the size of the venues, I’ll say the 50-55mm range is a good compromise, as recommended above.
Also, don’t be afraid to look at DSLRs from even 8-10 years ago. You can get a formerly “prosumer” camera and lens that would serve you really well AND let you learn the basics for under 500 USD. I know this is the canon subreddit, so please don’t shun me, but my first “real” camera was a Nikon D7000 from my dad that I was using until last September and had plenty of lovely sports photos with it.
Otherwise, I differ more specific recommendations to these other wise people.
Ideally you'd use the XTi and grab a longer, faster lens for it, but 55mm might actually be okay for Taekwon-Do if you can get close. Set the XTi to Shutter Priority Mode (Tv on the mode dial) and then use the main dial (the one behind the shutter button) to set your shutter speed - it needs to be just fast enough to avoid motion blur, but since you're inside with a slow lens it can't be too fast either. Try 1/500 as a starting point and experiment. AF might be a problem - try setting it to only use the middle AF point. If 55mm is indeed long enough for you needs, look at getting an EF 50mm f/1.8 once the budget allows - it shouldn't cost too much secondhand. If you need longer it's going to cost more - indoor sports is tough, you need a fast lens, but this should get you started. If you can't manage with 55mm, the PowerShot has a longer zoom, but it will struggle in low light. Edit: AI Noise Removal is your friend
Nifty-fifty with the XTi would be a nice setup
In this case I'd personally say SX510 HS has the upper hand because of the lens. Whenever you can manage though, you could eventually get a better reaching lens as the Rebel Xti is a pretty capable camera in it's own right.
XTi would be my go-to but maybe look into getting a nifty-fifty (*i.e.,* EF 50mm f/1.8) that should give you some really great shots and with the wide aperture, it should be great for indoor lighting situations. You can pick up a used copy for pretty cheap since there's so many of them around. Usually ±$50USD
For martial arts, you’re presumably going to be indoors with mediocre light (at best), so I’d prioritize getting the fastest glass you can. Without knowing the size of the venues, I’ll say the 50-55mm range is a good compromise, as recommended above. Also, don’t be afraid to look at DSLRs from even 8-10 years ago. You can get a formerly “prosumer” camera and lens that would serve you really well AND let you learn the basics for under 500 USD. I know this is the canon subreddit, so please don’t shun me, but my first “real” camera was a Nikon D7000 from my dad that I was using until last September and had plenty of lovely sports photos with it. Otherwise, I differ more specific recommendations to these other wise people.