So this is my system, I don't know if it will work for you so feel free to ignore, but it works for me.
Everything runs off my outlook inbox, using the 'tasks' functionality - I have my laptop with me at all times including in meetings which is fine where I work so I can add things in as a I go along, but if in your work you don't have a laptop or can't take it with you into meetings etc. then you might need to recreate it using a paper notebook.
So everything I need to do is either an email or a task - like you most of my work comes in on incoming emails but if I pick up actions in meetings or from phone calls/direct from colleagues or have regular scheduled tasks it is easy to create a new 'task' in outlook, you can set them to reoccur as well so e.g. 'expenses' once a month at the end of the month. I categorise them using the colours in outlook, so 'green' is things I need to action, 'yellow' is awaiting response/action from others, 'red' is to do with meetings/project groups, 'purple' is for referral/escalation to my manager etc, but you could use whatever makes sense to you. I then use the 'flags' to set a due date for action - within 24 hours if urgent, within 3 days if not urgent, within 7 days if I am awaiting input from someone else (so I remember to chase it up if haven't received the thing I am waiting for from them).
The crucial step is that I religiously set aside 30 minutes at the start of every single day to go through my inbox (you might need an hour if you have more than one inbox to go through, or do 30 mins at the start and end of the day) and categorise/flag anything that came in during the previous day/overnight, delete junk, file things needed for future reference (I have archive folders for this) and tidy up any completed or redundant tasks. From this I generate my 'job list' for the day, it's usually quite clear from the flags what is due today/tomorrow etc and outlook can generate a to do list for you - just remember to look ahead to the things which aren't due for a week but may need lots of work doing so you allow yourself enough time, you may want to break down bigger jobs into smaller tasks for this reason.
I then try not to pay too much attention to incoming mails during the day and focus on actually doing the tasks, obviously I keep an eye on it and respond to anything genuinely urgent but otherwise things can usually wait until my next 30 mins email session to be sorted. Sometimes I need to book in a longer 1-2 hour session to audit my full inbox and tidy everything up but I can usually fit this in on a Friday when things are quieter. I also usually try on a Friday afternoon to do a look ahead to next week and identify any busy days or if I am out of office for any days, as I may need to make an effort to get ahead on emails if I know I won't have much time the following day, so if I am on annual leave on Friday for example I may 'book out' time in my diary or set a reminder to myself to spend a few hours Thursday PM completing everything or at least apologising if I won't be able to come back to people until Monday...