Feeling the Eighth Note Subdivision
The first place to start with subdivisions of the beat is learning to feel the eighth note subdivision. Most music is in 4/4 time (see time signatures). That means 4 beats per bar/measure and each beat equals 1 quarter note. If we divide each of those quarter note beats in half, we get 8 eighth notes.
You need to learn to feel where those eighth note divisions fall and be able to consistently play them. This requires knowing where the beat is, knowing where the subdivisions are, and developing the technique to play them on your bass.
Playing Steady Eighth Notes
The first 8th note rhythmic skill to practice is playing a steady stream of eighth notes. The notes need to be spaced evenly. The 'ands' should fall right between the beats. You need to focus and practice a lot to achieve consistency. They can't feel unsure or wobbly. Beginners often think they are playing evenly when they aren’t. You might record yourself playing or ask others for feedback.
Playing the Off-Beats
Next, you need to learn to feel these eighth note subdivisions when you're not playing notes on the beat. This can be tricky for a lot of beginners. Again, get some feedback from other people, a teacher, or record yourself. A lot of beginning bassists don't realize they're playing the off-beats in the wrong place. These off-beats should fall exactly between the click of the metronome. Don't get turned around and start playing on the click. Pay attention. It'll groove when it's right. Count out loud - "One-and-Two-and-Three-and-Four-and" and pluck on the 'ands'. Or, count "doom-tek-doom-tek..." and play on the 'teks'. The counting will really help you when you're starting out. (See counting rhythms.)
Alternate Notes on the Beat and on the Off-beats
Once you can play steady 8th notes and play the off-beats independently, you can practice alternating one bar playing on the beat and one bar playing only the off-beats. (You'll find this on the exercises page for this lesson.) This will test whether you've got the concept or not. It should feel free and easy when you play it.
Hit Any 8th Note Subdivision with Confidence
Another thing to practice is being able to play any of the 8th note subdivisions by themselves. You want to be confident you know where each subdivision lands. Pick one and practice it by itself. On the examples page I show you by playing the 'and of beat 3'. You can do it with any 8th note, or any combination of 8th notes.
Practice and Apply Your Rhythmic Skills
Once you can play these 8th note rhythmic skills consistently with the metronome, try them without the metronome. Also, create your own basslines applying off-beats.
Don't forget, you can work on this anywhere. Feeling rhythm is internal; it's not in your fingers so much as it is in your mind. Your mind tells your fingers what to do just like it instructs your tongue to make certain sounds when you speak. Work on your internal sense of rhythm as often as you can. It doesn't matter what you can play if the rhythm isn't happening in your bass playing.