This is really just a list of the things that work for me. But possibly they will work for you too.
Being productive requires two main things: (1) An organised method for keeping track of, and prioritising, tasks, and (2) The motivation to actually do the thing supplied by that method. These two are somewhat related, in that a key motivator is seeing progress.
There are three main places that I keep track of tasks.
First, there is my email inbox. I *highly* recommend the "inbox zero" strategy of deleting processed emails in order to keep your inbox empty, or very close to that (say, <5 messages). You can later recover emails as necessary from the deleted items folder.* Inbox zero is great, but has one simple problem: Some messages need processing at a later date. Solution: I have a subfolder labelled "For next week". I check that folder at the start of each week and move things back into the main inbox as necessary. In short: I basically have 3 email folders: inbox, deleted, and for-next-week, and those folders could happily be relabelled "do now", "done", and "do later". (When times are really busy, I add a further complication to my email strategy, which is to only resolve emails received before I started that workday, so that my work for the day remains predictable.)
Second, there is my actual to-do list. This consists of a single small post-it note each day. Most of the time, the things I need to do are located across both my inbox and this list. This post-it goes in the bin at the end of each day, and I write a new one each day. The list stays fairly short - its got to be achievable in a day!
Third, there is my calendar. This is key. It obviously contains actual timed events like classes, meetings, etc. But it also contains things that I will do on certain days (e.g. "write grant proposal", "read Liam's work"). My first task each day is to copy tasks from today's date into my daily post-it, above. When I get new tasks from whatever source, I put them on a date on the calendar (often arbitrary). By putting things into the calendar like this, my to-do list at any given time consists only in things that can and will be done *today*.
That's it. With this system, I remember almost nothing: my list of tasks at any given time is always just a list for that day, making it much easier to keep track of, and prioritise, those tasks. And creating that list each day is just a matter of looking at my calendar for the day, alongside my inbox.
No doubt this system could be tweaked further - using the snooze function with emails instead of the "next week" folder; self-emailing tasks instead of using a post-it so that your inbox is an exhaustive to-do list for the day; and so on. But I find this works well as it is.
*An aside: The email client Thunderbird has an excellent search feature, way better than Outlook. As a result, I never archive emails into different folders; I have just one big searchable folder. I don't know how people manage using Outlook.