What would be the best expression to say that I have been rather busy and therefore struggling to find time for many important things such as work, school etc.
Something like: I have been trying to balance my time between school, work etc.
There are a few idioms that would express the idea of not having enough time for everything.
Firstly, you could use between to list the various activities that are taking up your time - Between school, work, and taking care of my children [...]
Secondly, you could simply say I have very little time...but that's boring. Idioms are a lot more fun.
I am/I have been run off my feet is my personal favourite. It carries the connotation of rushing from one task to the next, without any time to relax or decompress between them. There aren't enough hours in the day is another option, specifically indicating that you're struggling to find time for all of your responsibilities.
If you're trying to show that you, specifically, are in high demand, you could say there's only one of me, which indicates that you feel there are contradictory demands on your time and that you can't fill all of them.
So, for your specific question - how to say that you have been very busy and struggling to find time - I would suggest something like:
Between school, work, and astronaut training, I've been run off my feet lately.
The best single word I can think of is multitask:
2 : the performance of multiple tasks at one time
// The job requires a person who is good at multitasking.
In your example,
I have been struggling to multitask (between) school, work, etc.
I replaced trying with struggling to give a more definitive impression that you are not only trying to multitask the activities but finding it difficult to do so.
The use of between is optional. It sometimes comes after and sometimes doesn't.