I am planning to complete a Masters in Physics (thesis based research) at my undergraduate institution after being rejected during PhD applications. I feel like this will bolster my research experience and strengthen my research application. After this, I plan to apply to PhD programs, both at my home institution and at other schools. Do PhD admissions committees favor a Masters degree with research? Did I make the right decision in going for a Masters?
In my own field (neuroscience) I don't think a masters really counts for anything in US PhD admissions (but these decisions are all made by humans on an admissions committee; maybe one of those humans reviewing your application likes a masters degree). Research experience you gain while doing a masters is valuable, however, it just doesn't matter that it's part of a masters program rather than a job or anything else. Sometimes a masters is the best access a recent grad has to research, though.
Unlike what Buffy's answer implies, in my field having a masters doesn't change where you are on the PhD track - you'll have just as many years and courses as someone coming in with a BS.
I can't say whether you made the right choice or if it will work out for you - no one can - but if you can gain experience in research and work with people that can write you strong letters of recommendation then this choice will increase your chances of PhD admissions. Whether it will get you into the program you want (or any program) can never be guaranteed, though.
This is hard to answer, but I have some advice that is too long for a comment. In general, in the US, there is unlikely to be much preference for someone with a MS vs someone with a BS for graduate admissions. Coming with only a BS gives you a longer time frame to get ready to pass comprehensives before you start dissertation research. A MS candidate will still need to pass those exams and may be ready at entry or not.
But among those with a MS, having research experience is probably a plus in many fields, though it might narrow your possibilities since it has narrowed your focus.
But the bigger question is why you were denied entry already. Perhaps your record is poor. Perhaps you applied to too few schools. Perhaps you applied only to very selective programs with a lot of competition.
If your own school rejected you for doctoral studies there is probably something lacking that should be directly addressed.
But, since your current institution has a doctoral program in your field, you have an opportunity to get serious and specific answers to all of your questions by speaking with some trusted faculty member (or two), who are in a position to give you career advice.
The other question you need to answer concerns you long term career goals. Will a BS alone or a BS + MS give you the path you want, assuming you still don't get in to a doctoral program? Think about what your career trajectory might be with all these options. For academia you need a doctorate, of course. For industry, not so much.