Just a recommendation: Tai Chi's Silk Reeling (more instructional video here).
Regardless of what style you do, I find this exercise to be beneficial for both fighting form and conditioning. Firstly, the movements are said to summarise what the arms do in every single fighting style. Some styles may do it in a more contracted way*, others may do it in a more expanded way**, but it's still fundamentally the same movement. Secondly, by moving the arms (and body) in multiple directions like this - and you can hold hand weights or strap on wrist weights while doing this exercise - it targets all the muscles in your arms and shoulders, and not just one isolated muscle or group of muscles as you would if you were doing say arm curls (which primarily focuses on the biceps).
But either way, I find it useful to practice forms with weights.Usually with strap on wrist and ankle weights, but I sometimes practice holding onto hand weights. There's nothing wrong with doing regular weights, but this just saves time - you get to practice your martial arts form and do weights at the same time.
Also, practising your form slowly with weights on helps IMO. Slow form practice is good because it allows you to focus on your techniques (which you can often miss by moving quickly) and of course, with weights on you feel the burn more as your muscles have to carry the weights as you move through different positions slowly (whereas when you move quickly, momentum does half the work for you). It also helps your posture and balance too (especially when you're slowly extending and retracting your kicks and holding them with ankle weights strapped on)
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*Note that the movement in 0:18 of this Shotokan Karate video is essentially the same as in 0:18 of the first Silk Reeling video link posted above, only more contracted.
**The movements seen in this Aikido demonstration are basically the same as silk reeling, but more expanded - particularly obvious/visible in 1:09