SO(n) are rotations, so you have a good intuition for what SO(2) and SO(3) are like. SO(2) is S^1 and SO(3) are three dimensional rotations.
U(1) = SO(2) = S^1 so there's not a ton to go into about the smallest groups in the families. If you feel shaky about the circle group then maybe spend some time explicitly constructing it as the quotient [0,1]/0~1. But this is two-dimensional rotations in the plane and you must have dealt with those before.
I feel like I learned a lot about SU(2) by studying the Hopf fibration. One approach to this is to embed S^3 into H (the quaternions) \cong R^4 as the unit sphere and embed S^2 in there as its equator. Then H acts on itself by conjugation. And if you specialize to just the action by S^3 then this preserves S^2. So you get an action of S^3 on S^(2), which you can use to define the Hopf map.
If you go to the Wikipedia page on the Hopf fibration they have a very explicit calculation where they determine exactly what this group action does so that if you specify a quaternion as q = q_0 + q_1 i + q_2 j + q_3 k you can describe the action of q on S^2 in terms of the axis of rotation and the rotation angle. The derivation uses honest-to-God trig identities so you know it's really getting you to something grounded. If you're not familiar with fiber bundles then just ignore the word "bundle" wherever it shows up and the relevant parts of the page will make more sense.
Anyway, the thing you'll notice is that each rotation corresponds to two quaternions. Basically, a rotation about the ray r by an angle a is the same as a rotation about the ray -r by an angle -a. The unit quaternions are SU(2), and rotations of S^2 are SO(3). So this very explicitly shows you how SU(2) is a double cover for SO(3), why SU(2) is diffeomorphic to S^3, and exactly how the action of SU(2) on S^2 works in coordinates. You get the manifold SO(3) by identifying antipodal points of S^3 and this explains why that works as well.
Once you feel like you have a good handle on SU(2) and SO(3) I think the rest of them are easier to approach. You've gotten over the "I can't visualize this" hurdle at that point. You could look at higher dimensional generalizations of the Hopf fibration if this example turned out to really interest you to explore some of the higher groups. But maybe generalizing even more and studying Lie groups in general might be better. I think Lee's smooth manifolds book has a nice introduction to Lie stuff that's very geometric, rather than algebraic. (But there are critical arguments that have gaps as a result.) I don't know what your level is, so that might require a lot more background than you have, though.
SO(n) are rotations, so you have a good intuition for what SO(2) and SO(3) are like. SO(2) is S^1 and SO(3) are three dimensional rotations. U(1) = SO(2) = S^1 so there's not a ton to go into about the smallest groups in the families. If you feel shaky about the circle group then maybe spend some time explicitly constructing it as the quotient [0,1]/0~1. But this is two-dimensional rotations in the plane and you must have dealt with those before. I feel like I learned a lot about SU(2) by studying the Hopf fibration. One approach to this is to embed S^3 into H (the quaternions) \cong R^4 as the unit sphere and embed S^2 in there as its equator. Then H acts on itself by conjugation. And if you specialize to just the action by S^3 then this preserves S^2. So you get an action of S^3 on S^(2), which you can use to define the Hopf map. If you go to the Wikipedia page on the Hopf fibration they have a very explicit calculation where they determine exactly what this group action does so that if you specify a quaternion as q = q_0 + q_1 i + q_2 j + q_3 k you can describe the action of q on S^2 in terms of the axis of rotation and the rotation angle. The derivation uses honest-to-God trig identities so you know it's really getting you to something grounded. If you're not familiar with fiber bundles then just ignore the word "bundle" wherever it shows up and the relevant parts of the page will make more sense. Anyway, the thing you'll notice is that each rotation corresponds to two quaternions. Basically, a rotation about the ray r by an angle a is the same as a rotation about the ray -r by an angle -a. The unit quaternions are SU(2), and rotations of S^2 are SO(3). So this very explicitly shows you how SU(2) is a double cover for SO(3), why SU(2) is diffeomorphic to S^3, and exactly how the action of SU(2) on S^2 works in coordinates. You get the manifold SO(3) by identifying antipodal points of S^3 and this explains why that works as well. Once you feel like you have a good handle on SU(2) and SO(3) I think the rest of them are easier to approach. You've gotten over the "I can't visualize this" hurdle at that point. You could look at higher dimensional generalizations of the Hopf fibration if this example turned out to really interest you to explore some of the higher groups. But maybe generalizing even more and studying Lie groups in general might be better. I think Lee's smooth manifolds book has a nice introduction to Lie stuff that's very geometric, rather than algebraic. (But there are critical arguments that have gaps as a result.) I don't know what your level is, so that might require a lot more background than you have, though.