When I get knocked down, I become completely vulnerable until I get on my feet and have a chance to jump, block, or do a fight move. I'm hoping this is because I'm inexperienced and not because there really isn't anything you can do but hope the other fighter backs off and gives you an opportunity to get up and do a move, block, or jump. But Zangief never gives me a chance. So is there anything I can do?
Characters who are knocked down are invulnerable to nearly all attacks. When standing up after being knocked down, you always have a chance to block an incoming attack, although it might be difficult to do correctly if the opponent crosses you up.
After rising up, there is also a very short time frame (2 frames) before a character can be thrown. Against Zangief and throws in general, if you use a reversal move with start-up invincibility (dragon punches, etc.), it is possible to hit them if they try to throw you. You can do this against regular attacks as well, but keep in mind that reversals put you in a bad position if blocked.
In other words, no, you can't do anything while being knocked down, but you always have a chance to block or evade their attack after getting up. It is not always easy, and specific situations might require a bit of practice. Knocking the opponent down is a very advantageous situation exactly because it severely limits their options.
However, if you are knocked down by an attack other than a throw, sweep, super, ultra, or certain specials, it is possible to decrease the knockdown time drastically by doing a tech, also called a quick stand or quick rise. This can be done by hitting two attack buttons or tapping the joystick down just as you are about to hit the ground. While teching makes it more difficult to follow up in most situations, it does not entirely remove the problem related to being knocked down; you still need to anticipate and deal with the opponent's next move.
Also, while it might not be relevant to you, a new mechanic added in the Ultra version allows a player to very slightly delay their wakeup from hard (untechable) knockdown. It is also done by hitting two attack buttons or holding the joystick down, and can be used to throw off an opponent's timing. This can be useful if the opponent tries to time a throw on wakeup, for example.
The character Cody has a move you CAN do when knocked down. It pushes your opponent back and deals no damage, and can be blocked, but at least when it works it gives you a little breathing room. Otherwise, getting knocked down becomes a rock/paper/scissors battle to get back up and turn the tide of the fight.