With a little bit of knowledge a set of 2m resistance bands and James Grage's videos and channel (https://www.youtube.com/watch?v=PfSoC-IZMOI for a start) will set you up for cheap and very little space muscle training in the home with really minimum friction to starting and potential for injury. A lot of people go for weights but the muscles don't care resistance is resistance and bands can do things weights can't. When on your own at home you can push to failure safely with bands as you can just let go to remove all tension.
For cardio its mostly about a machine and bikes tend to be smaller and cheaper than most of the others but a jogger or a stepper will get you there too. Really depends on how much cost you want to lay out and how much space you want to assign to working out.
What I found was really important was having something I wanted to do with this fitness and using that as a focus and reason for being there. I also watched TV while doing cardio and that guaranteed I would get 40+ minutes of a TV episode time on the cardio side of things without getting too bored.