I haven't used to run much, but recently I started working out and it includes running also. I noticed that after I run about 400-500 m, my legs (both) start to hurt in the area of soleus muscle, on the internal side of the lower leg, about midway from the foot to the knee, pretty localized close to the bone. If I continue running or jumping, it becomes worse until it's quite intense so I can't run anymore. If I leave it alone, it passes in a couple of days and never reappears unless I run again or do a lot of jumping. Any ideas why it's happening, is it because I'm doing something wrong (wrong shoes? wrong movements? weak muscles?) and if anything can be done to improve it?
Pain in the lower leg muscles and tendons is very common among beginners who start to run. The reason is that your muscles are not yet adapted to running, meaning there are specific muscles which are either too short or too weak in relation to the others. This imbalance causes excess tension on these muscles and can result in injury if you keep running anyway.
In my own experience (I've had the same problems), these issues will disappear completely if you keep trying to run until you feel the pain, rest for a couple of days, try again, and so on. After a few weeks the pain will disappear for good. Your muscles will adapt to the motions automatically. Just make sure you don't injure them, do not ignore the pain, and don't wait until it becomes literally impossible to continue. Once the pain can be felt clearly, you have given your muscles more than enough of an impulse to adapt while you rest.
If you want to speed up the process, there are lots of strengthening and stretching exercises you could do to build muscles selectively.
Here's a good overview of different kinds of leg pain from running, including yours, and what you can do about them: