Although there is no replacement for an actual drum set to learn on, there are a multitude of exercises and rudiments one can work on without a drum set.
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Step 1: Buy a Drum Pad and Drumsticks
There are so many types of practice pads available today. I like the ones that are around 12 inches and/or fit nicely on a snare drum. You can also place it on a snare stand or anything in your place that holds it. However, you want it at a comfortable height.
Buy a pair of drumsticks. Any size will do. If you can't decide, you can go with 5A or 5B - They are very popular.
Step 2: Finding Two Sound Sources on the Pad
Now we want to find two different sounds on the pad - this can help us simulate a drum set. I usually hit the upper side of the pad (the edge) with my riding hand. (If you are a righty, your right hand will go up and hit that area.)
And then we can hit the center with the left hand (your backbeat hand).
Now slowly try this hand coordination pattern:
"Right Right Both Right; Right Right Both Right" - which you can also think of as "1 an 2 an 3 an 4 an". Your 2 & 4 is known as your backbeat.
In essence, your right hand is mimicking the hi-hat or ride cymbal and your left is mimicking the snare drum.
Step 3: Playing Your Feet on the Ground
Next, you want to get comfortable playing your right foot on beats 1 and 3. Practice lifting your foot up and down to get used to this movement.
You will have a complete beat when you add the foot on 1 and 3 with the aforementioned hand pattern.
Things will make a lot of sense if you get a book to explain this and much more. My book Drumopedia is a drum book for beginners that breaks down things to small kernels, and goes over some reading, coordination, beats, fills, notation and more.
Step 4: If You Don't Have a Pad and Drumsticks
Now, let's say you do not have a pad and drumsticks. You can try those patterns on your knees with your hands or on anything that is audible.
Step 5: Learn Some Basic Rudiments
In drumming, a group called the NARD (National Association of Rudimental Drummers) created a standard of stickings decades ago, and then this list was updated by PAS (Percussive Arts Society).
Here are some important rudiments to practice with your drumsticks on the pad, or your hands on your knees:
1) Single Strokes: This is when you alternate right to left.
2) Double Strokes: Play two right hands hits, then two left hand.
3) Flam: Hold one hand low and one high, and hit with your lower hand slightly before the higher hand. This produces a thicker, wider sound.
4) Paradiddle: This a combination of single and double strokes: RLRR LRLL
With all of these, keep the strokes evenly-spaced.