When I have some songs to learn I'll put them in a playlist and listen to them on repeat for a couple of commutes. Then, I'll sit down at the computer with headphones on and open up a new document (I use Google Drive because it's so handy and I can have my music anywhere).
I'll listen to it through once again, naming sections, and writing how many bars each section has. I'll listen to it once more, checking that working, tapping and counting along. At this point I'm not worried about what is played in each section, just the structure and counting of the song.
Once I have that skeleton to work with, I'll go through each section of the song and make notes on what to play. Since most songs I play are relatively simple, this is normally a note on which cymbal to ride and how fast, and where the bass and snare will go in each bar. I'll make notes about which bars have interesting fills in, and accents on cymbals and such, but I don't go crazy with details. Most of the later sections just have notes saying "repeat as above Chorus" or such. This rarely gets longer than a single sheet of A4.
I'll then play it through beginning to end, print off the sheet, and take it to practice. Then about the first 3-5 times I play with the band I'll use the sheet, and then I'll have internalised it enough to play it confidently without the sheet. I'll still be counting in my head, but this whole approach is exactly why my band members rely on me to remind them of the structure of a song as we practice it.