Please see a doctor if you're at the point of problems with balance, weakness/no strength and unsteady feet as described. Might be a nutritional issue (deficiency of certain mineral/macronutrient) or something else that warrants qualified medical expertise.
If your doctor clears you and you're good, it might just be not enough caloric intake before working out (eat a banana before lifting). Track your calories on myfitnesspal and figure out how much your BMR using this site to get an idea of how much you should be eating in a day. Chances are that feeling of weakness is a lack of calories (assuming you've been cleared by a doctor of any health issues). Understanding the concepts of nutrition is really important because your body is essentially built in the kitchen.
Start with easy weights or bodyweight exercises and work your way up (with a trainer or someone experienced if possible). I posted this as a suggestion before for someone seeking training advice and suggest the same to you:
If you're at the gym, I would highly recommend you start lifting if your PT doesn't have you lifting weights yet. Use light weights, search up form vids online and do the lifts properly. Get used to these lifts. This is not to say ignore bodyweight exercises and cardio because those are important as well, but since you are paying for a gym, might as well capitalize right? You mentioned you had a weak back so things like deadlifts/rows (crucial to have good form) and lat pulldowns, cable rows will help build up your back. Start getting an idea of the different exercises and which parts of your body they hit. For example, you could do a standard bodybuilding style workout program where you hit different body parts every day with 10 to 12 repetitions per set. Or maybe you want to start lifting as a sport/strength gaining regimen and might want to check out stronglifts 5x5. There are lots of programs and it's cool to experiment but you want to pick something and stay consistent. Don't be the kid in the candy factory after a month or two of being comfortable with the gym. Pick a program and stick with it for the goals you set for yourself.