How you run varies a lot by country.

In the US, you start by filing papers to do it. Typically a combination of a fee and collecting signatures. Municipal government tends to have a lot of power over things like what commute mode is favored and local building codes, so I recommend it for first-time candidates. You'll need to raise enough money to help get the word out, spend a lot of time visiting community groups and talking with people, and ideally recruit a few volunteers to help you win a primary.

You'd be surprised how easy it is. Many political parties will allow anyone to run for a seat if no one else is already doing it. A lot of the smaller parties will even actively seek out good candidates.

There's vetting of course, but so long as you're not a terrible person and haven't said anything publicly that's opposed to the party's platform, it's really not too hard.

The hard part is when it comes to the larger parties that have established (often terrible) candidates. The party knows them, so you have to put in the time to get known as well and then run against the sitting candidate when the time comes.

It does happen though. AOC in the US is a great example. She unseated a democrat that the party actually liked, and she did it with an army of new members she got signed up.