Feel the Pulse
A large part of your rhythmic abilities depends on feeling a steady, even pulse. After clapping with the metronome, another way to develop it is to practice playing a steady stream of evenly spaced notes on the bass. You need to connect your inner sense of the pulse to your bass playing technique.
It’s not super exciting to practice, but it helps a lot. If you can’t manage to play a stream of notes steadily, any other rhythms are going to be shaky and off. No one wants a bassist with bad timing.
Next I'll suggest how to work on this and then I'll give you some audio example exercises...
Practicing the Pulse with the Metronome
To develop your steady pulse, you can practice plucking on every click of the metronome. (You can use my online metronome if you don’t have one.) Start slowly between 50 and 60 bpm. Then gradually increase the tempo until you’ve reached the fastest tempo on your metronome (usually around 200 bpm).
Playing on every click will insure you are playing the notes evenly. Most beginning students think they are playing notes evenly when they are not. You can record yourself or ask another musician or your teacher if your notes sound evenly spaced.
You should also focus on playing the notes with a consistent volume. One finger or pick stroke should not be louder than the other. It takes time and practice to gain control. Again, you're not a robot so don't expect absolute perfection all of the time. With work, you can get pretty consistent however.
Then, Without the Metronome
Next, you’ll want to practice playing that pulse without a metronome. After all, that’s what you’ll be doing when you play with others. You may even want to think of yourself as everyone else’s metronome. Practice the same steady pulse. Pay attention to your tempo. Try to not speed up or slow down. It’s not easy at first. Record yourself.
You'll find that when you play with others some people tend to rush or drag. You have to learn to not let their bad time influence yours. You don't want your 70bpm ballad accidentally speeding up to 100! This is where having a strong internal sense of rhythm is going to help you. It's best when everyone has it, but a lot of times others will lean on you.
As simple and boring as this stuff is to practice, it will pay off in everything you ever play. Work on it a few minutes a day until you can keep it going for long stretches of time without making any mistakes. Eventually you'll get to where this doesn't require any of your attention when you play. Good luck and practice diligently!
There are a couple of examples of playing on each click on the exercise page to make sure you know what I mean.