None of us knows how flexible you are, how strong you are, how good your stamina is, etc.. We can't really answer whether you need any weight training (and any kind of cross training) or not. Indiscriminate advice is all we can really offer you, based on the little we know about you. You said you need or want weight training, so there's nothing more to discuss about it, IMO.Wow, I'm coming across very differently from what my original intentions were (thanks for pointing that out!) and for that I apologize.
I most definitely do not want to dismiss ballet training. I wrote that I'm reading up on ballet training to get some ideas on foot and ankle strengthening. What I said in the previous post was that I thought Pilates was "not for me," because I am not at the level of fitness of a ballet dancer or presumably a different dancer that has benefited from Pilates. I'm not against Pilates, yoga, and running, I'm against people recommending them indiscriminately to everyone, without any sort of caveats.
I also don't want to dismiss suggestions from people. I am trying to move the conversation to a more specific level. I've been dancing for a while, so I've heard a lot of these suggestions and tried most of them. Now I'd like to understand the why's. Why is yoga helping people and what exactly about yoga is it that they are benefiting from (ok, stretching, but what exactly is it important to stretch?). What muscles is Pilates strengthening that are helping dancers, so I can target those muscles in a different way that perhaps won't hurt my lower back (sure, the "core", but what exactly does that mean?). I guess I'm curious if people have dug into this.
In any case, it's my personal opinion that your focus should be more on what the issues you are having with your dance, and how to improve them (which would include things specific to tango technique, instead of just cross training).
Example, you alluded to some issues with pivoting. It might be more productive to ask about technique about whatever move involving pivots (like back ochos), that you are having problems with.
While cross training could help some, working on your tango technique will likely help a lot more.
Now with that being said, you asked about core. Your "core" (or core muscles) are the muscles around your abdomen and back. These muscles help with your stability and balance (among other things). An awful lot of people think pilates is an excellent way to strengthen core muscles. However, like I've already said, we don't really know if you need to strengthen your core or not.