Beat & Rhythm are NOT the same. This video helps show the difference using spoken poetry: the beat is the jumps, the words are the rhythm. (You could try this same activity at home, though for little kids, just have fun keeping the beat & varying the volume. For older kids, try to figure out some (or all) of the rhythms-- they aren't straight-forward!)
So far I've listed several games for reinforcing the idea of steady beat, but haven't touched rhythm yet. Beat & pitch are the two main objectives you should always be reinforcing with your little ones, right down to gently patting the steady beat of hymns in church or stomping to the beat of a radio tune (then ask whether it was 'fast'-allegro or 'slow'-adagio or medium- andante. See a full list of musical terms for speed- tempo- here). However even young kids can enjoy exploring basic rhythms. Use this process to teach quarter notes, eighth notes & quarter rests. Go at the pace of your children-- introduce the idea, enjoy it, and leave it. You could spend a week or a day or 10 minutes or a month on each step:
1. introduce the idea of stops (hands held out, palms up, in an "I don't know" kind of position), slows (a clap- keep your hands clapped together for a few seconds) and quick-quicks (claps twice as fast as 'slows')
2. have your children echo 3-beat patterns, such as "stop, slow, stop" or "slow, quick-quick, quick-quick"
3. label your 3-beat pattern a "rhythm." Use the terms rhythm & pattern together often in describing what you are doing.
4. allow your children to take turns clapping a 3-beat rhythm with you echoing them. You can have them speak a rhythm without clapping first, then show them how to clap it and have them clap it with you if they have trouble speaking & clapping simultaneously.
5. explain to children that slows, stops & quick-quicks can be written down, either with
- green rectangles, yellow squares & red squares
- joined eighth notes, quarter notes & quarter rests
- combo of two
6. Set out a 3-beat pattern (literally lay out the rectangles/quarter notes), perform & have them echo. Let them compose, perform & echo.
7. Introduce the idea of a measure by making a measure sheet that has 3 (or 4) empty boxes into which the quick-quick, slows & stops can fit, with a modified time signature (top number- # of empty boxes, bottom either a drawn quarter note or a yellow box) and measure bar.
8. Repeat process with 4 beat patterns.
9. Down the road, create sheets with 2 measures (each containing either 3 or 4 boxes & modified time signature as above) introducing the idea of an end bar.
**Can experiment with adding in words or pitches (could add in 'sol' 'la' and 'mi') for older children.