I'm currently using DaVinci Resolve to subtitle my videos but I realize now that it has no support for people speaking with multi track subtitles. I want to color code subtitles for different speakers and perhaps move the subs closer to the speaker depending on who's speaking. What other software should I be using instead? Does premiere have a good workflow for this usecase?
Resolve does support multi-track subtitles. To add a second subtitle track, right-click in the track header and select "add subtitle track." This is used for language localization, so that you can deliver different caption versions for the same program. Resolve is advanced in this respect, with support for updates via supplemental IMF and DCP, so that film and broadcast studios don't need to re-export entire programs whenever they need to make minor corrections.
You won't be able to use two tracks at the same time in any software, because you can't display two languages of captions at the same time within current broadcast television standards. It's the same reason you can't choose English and Spanish on your television at the same time.
If all you want to do is control the appearance of two different styles of text, use the "Title" category of tools, located in the effects library, not the subtitle track. I'd recommend "Text+" over the legacy "Text" which doesn't offer any advantages. Define the settings you want for each style on independent titles on the timeline, then drag a copy of each look from the timeline to the media pool. Use these as your templates/presets, it helps to rename them accordingly.
If you need to alter the appearance of a batch of titles at the same time, you can copy the source style via cmd/ctrl-C and then "paste attributes" to multiple destination titles via alt/opt-V.