- ウェイトを付ける/着ける is the safest, common choice in this case. Both for ankle and wrist weights.
- ウェイトを巻く is fine if you want to use a more specific verb.
- ウェイトを装着する is also OK, but may sound too technical in casual conversations.
- ウェイトを締める is acceptable, but sounds a bit off. It's usually specifically used for belts.
- ウェイトを着る: wrong.
- ウェイトを履く: wrong because it's only for
shoes and socksshoes and, socks, trousers, underpants, etc.
