@LittleRebelGirl I understand about wanting to lose the weight first, but honestly you might be better off eating a bit more and getting stronger. If you can add muscle and up your BMR/TDEE you'll find the weight loss much easier in the long run. You build a bit of muscle for a while, then cut your calories again (but not as low as 1290) and you'll lose weight more quickly and easily
If it helps, I'm about the same height as you and 60kg, with quite a lot of muscle for my frame. Before doing weights my maintenance was around 1400 calories. Currently I think it's around 1800-1900 (I have a sedentary job but I do do sports as well as strength training) That gives me a lot more wriggle room.
Compound movement examples if you want to look then up -
Squats (any variety, but usually not advisable on a smith machine)
Deadlift (again, lots of varieties)
Bench press
Overhead press
Rows (machines are fine for these, eg seated rows)
Pull ups (assisted on machine, or lat pulldown on a machine)
Hip thrust
If you can't afford a PT, then maybe pick one exercise, look up some videos on YouTube and try it a few times in the gym before picking another one. It's important to practice movements with lighter weights to feel confident with it at first, that's something you can start doing any time. Also while there's a million YouTube videos for the above, if you want to find somewhere to start, I find Mindpump have a decent amount of free resources, and quite a good podcast too (don't always like their general chatter but the fitness advice is good! They tend to be more realistic than the twenty year old influencers that dominate a lot of fitness spaces)