It is pretty difficult to have a strong half guard but be bad at Jiu Jitsu. Understanding top and bottom half guard is a heavy task. There are lots of moving parts to the position. That is one of the reasons we introduce it early.
There are many ways to play half guard. I prefer the approach outlined here because I find it safe and when set up correctly, the top person does a fair amount of the work for you. Setting it up correctly in this case means don't rush the technique and pick your starting point carefully. That means getting so good at the first portion of the move, "protecting the shell", that you are able to choose when you enter the move and surprise your partner.
When you are playing half guard you want to establish a position where you are laying on your side with your back almost perpendicular to the mat. I use my limbs to keep you from controlling me, (by getting an underhook, head control, or past my knees). In the first example you will see me parrying for inside control. My right hand prevents head control. My left arm frames and plays a supportive role for my top leg, helping to keep it from getting smashed. If the leg does get smashed I circle out and reestablish the position. I should be using my right knee to prevent my partner from getting past my leg. If he does get past my knee my right elbow will frame inside their thigh and help me reestablish the shell. In the first example you see me effectively protect the shell for a bit, but when Chris gets past my defenses he is able to establish control and flatten me out.
If I am able to effectively protect my shell I can begin to look for openings to establish a more secure half guard. There is a bit of danger in this transition. If you don't have timing on your side, or use space effectively, it is very possible for your partner to flatten you out and crush your hopes of half guard stardom. Timing and clean technique are what make this work, not a good clean and jerk. You are essentially trying to dive into hole with this set up and move low on your partner, securely attaching yourself to their leg.
Try to time the move so that your partner is driving into you when you dive into that hole. The hole is established by straightening out your legs and allowing your hands to follow. Do not just straighten out your arms and legs, jack-knife your body and DIVE in a far as you can go. You are still on your side but you are trying to crunch in and activate those summer abs. When you are as far as you can go use the space behind you to lift your partner up. All of the pushing and pulling is done with the insides of your thighs/arms, not your hands. Rock back and straighten your arms and legs straight up. This is a very foreign method of moving another person, so be patient with yourself and others while everyone is learning it.
You do not want to do all of the work of getting into proper lifting position to just lift your partner up and set them back down in the same place they started from. The aim of the lift is to move yourself underneath your partner. Remember that you are starting on your side, lets say your right side. You are then rocking towards your left side, (but stopping at the halfway mark). You want to leave your partner there while you safely return to your right side. Your left bicep/forearm is the E-brake that keeps your partner from following you back. Keep your left elbow pointed skyward as you rock back your right side and use the momentum to pop up onto your right elbow.
When this is done well you can use a common follow up and pop out the side to establish top turtle position. A similar follow up is to secure back control.
Often times getting out of the bottom position isn't easy. When your partner is attaching themselves to you and making it difficult to go to the back you can often follow up with a sweep commonly called old school. To perform this sweep you need to effectively secure their far foot. This is done by grabbing the toes. Notice that my first contact with the foot isn't a grab, but a kind of karate chop motion to scoop the toes up and hand them off to the other hand. You want to grab the toes and not the ankle. An ankle grip is easy to kick out of, while grabbing the bottom 1/5 of the foot makes it tough for your partner to get away.
Once the far-side foot is secure, block off the rest of that leg by reaching under their shin with your bottom side arm and grabbing by the knee. From there you want to pull back with your top leg, carefully applying a little torque to your partners leg. From there you push your belly in and log roll over their leg. It is a fairly effortless sweep when set up correctly. If you feel like you are muscling it you should look at your technique until you are smoothly knocking over a non-resisting opponent.
Doing the move in class doesn't give you a strong working knowledge of the technique. You need to spend a little time working on the move outside of class. Once you have the set up and options down roughly you should positional spar with your training partners to get the feel for how it works real time. When you find areas of the technique that are falling apart, go back to drilling just that portion of the move. It takes some practice to get a decent half guard game, but like I said it is very difficult to be good at half guard and not good at jiu jitsu. The position, or the opportunity to force the position, comes up in most matches.
Happy training, see everyone in class!