How to take session notes as a DM without missing the game
Nobody warns you that notes are half the work of a long campaign. You can prep a great session and still lose the name of the NPC you improvised halfway through it. These are the habits that help me keep up without spending the game staring at a keyboard.
Write down names first
During the session, names are the one thing you really have to catch. You'll remember what happened for a day or two, but the name you gave the ferryman on the spot is gone by bedtime. Keep a short list open and add every name you make up, with a few words about who they are.
Keep a one-line log
Next to the names, write a single line whenever something changes, like the party taking a job or someone making a promise they'll regret. Keep each line short enough to write while a player is still talking.
Ten minutes after the game
The best time to write the recap is straight after everyone logs off, while it's still fresh. Go down your log and turn each line into a sentence or two. Finish with the threads that are still open, because that list is most of your prep for next time.
Let a player help
Plenty of groups have one player who enjoys keeping notes. Let them. Ask them to post a recap in your Discord before the next game and read it out as the opener. You get a second memory, and the table gets a reminder of where they left off.
Keep everything in one place
Notes split between a doc, your phone and the back of a character sheet might as well not exist. Pick one place and link things together as you go, so the ferryman's page mentions the job he gave the party. Once a campaign passes a dozen sessions, a wiki holds up much better than one long document.
Or let the session write them
If you play on Discord, you can skip most of this. SCRY records the session and writes the recap and the wiki pages afterwards, including the ferryman whose name you made up on the spot. You still read it and fix what needs fixing, but you don't have to write it.
Whatever you use, the goal stays the same. When a player asks about something from six sessions ago, you want to find it before the table gets restless.