Congrats on starting getting into fitness, among other things it does help with confidence, self-image, and mood.
You got started with a P.T which is a really good idea as she can show you various exercises and teach you the correct form - very important!
No matter how long you've hired her, only 4 sessions with her would help you get started.
Ask her to get you familiar with machines as well, and how set them up properly.
Then ask her to make you a program you can do on your own for 2-3 months.
After that you'll have enough knowledge to know what you like / don't like and find a sound program on the web you can try, or just make up your own.
Don't work out everyday now.
If you're a new starter, your body needs to get used to exercise, adapt, and needs rest to recover and get stronger.
Always keep at the minimum 1 day/w of rest. Right now I think 2-3 days/rest would be better.
I suggest you do 2-3 weights sessions / week, and only if you fancy it and don't feel tired, maybe 1 cardio session (what you like really, walking, running, try as many things as possible and find your favourite ones).
Or maybe 2 sessions weights + 2 cardio sessions, up to you really.
You can do weights 1 day and then cardio the other day, or if you prefer do weights and cardio on the same session, in which case it's usually recommended to do weights first.
For instance in a week:
day 1 - weight - full body - 5 exercises, 1 per body part
day 2 - rest
day 3 - weight - full body - 5 exercises, 1 per body part
day 4 - cardio
day 5 - weight - full body - 5 exercises, 1 per body part
day 6 - rest
day 7 - rest
day 1 - weight - legs - 6 exercises
day 2 - rest
day 3 - weight - chest - 6 exercises + cardio
day 4 - rest
day 5 - weight - arms - 6 exercises
day 6 - rest
day 7 - weight - back - 6 exercises + cardio
Find what you like best and fits into your schedule and your life.
Keep a log to track your progress. Make sure you take things up as you get stronger, but do so slowly
to avoid niggles and injuries. Keep things varied as well.
HIIT is effective but it is very demanding, even the fit dread a good HIIT session. Get into slowly.
Listen to your body : if you're genuinely tired, the best option is too have a short easy session, or just call it a day and go home.
You can always take that up later when you get fitter. If you stick to it, I bet you will
Food : don't worry about macros now (but just so you know - a balanced vegan diet that includes tofu/soy milk/beans on a daily basis might be around 15% prot, 65% carbs, 20% fat).
Make sure you cook your own unprocessed food - legumes, veggies, fruits, seeds, nuts, grains - with the right portion sizes and things will take care of themselves.
Say 1 portion fits into the palm of your hand, a meal could be : 1 portion beans, 1 portion rice, 2 portion veggies + small bit of fat for cooking.
If you want to lose weight, give even more place on your plate to veggies.
If you feel hungry between meals, snack on fruit and nuts, but careful as nuts brings plenty of calories.
Listen to your appetite.
To monitor your progress: take pictures and monitor your weight once a week or every fortnight to see how you're doing.
Don't be too hung up on numbers, looking and feeling better does not necessarily mean weighting less.
Keep your fitness varied and challenging, and have fun
If you do that with the right eating habits, you'll be pleased with your results.