A quarter note is one beat. Cut it in half and you have two eighth notes, counted "1 &" — the number on the beat, the ampersand on the upbeat. Eighth notes are the first thing you really subdivide, and they unlock most of the grooves you'll meet in the first year.
Each exercise below stays on the snare so you can focus entirely on the counting. Say the count out loud — literally out loud. Internal counting comes later; right now your mouth is part of your timing.
Exercises
1 — Quarter Notes (Just the Beat)
One snare hit per beat. Say 1, 2, 3, 4 out loud, in time. Each spoken number lands exactly on a stroke.
2 — Eighth Notes
Twice as many strokes per bar. Now say 1 & 2 & 3 & 4 &. The numbers still land on the same beats — the &s slot exactly halfway between.
3 — Mixing Quarters and Eighths
Pattern: quarter, two eighths, quarter, two eighths. Counts: 1, 2 &, 3, 4 &. Keep saying every count out loud — even when your hand isn't striking on it (e.g. the & of 1 is silent here, but you should still say it).
4 — Eighths with Rests on the Beat
Strokes only on the &s — the upbeats. Beat 1 is silent, but you must still say it: (1) &, (2) &, (3) &, (4) &. Hitting only the offbeats while feeling the beats internally is one of the hardest first skills.