I have joined a political party 2-3 years ago and I can give you my own account of things. Mind you I joined the political youth of the party, which is generally more active.
So, the first step you can take is actually join a political party. Make sure the one you join is actually the one most ideologically close to your thoughts (sometimes people think their own ideology is the one of party X, but actually it's of party Z. This happens mostly due to political marketing).
As someone said, politics is Local and you will probably have to join the section of your county/district. The working methods vary greatly from county to county and district to district. Expect great rivalry.
The party should have several events per month, and you should be able to join them, listen, debate, make questions. Most importantly be seen and be heard.
Be prepared to navigate a jealous sea of beta mediocrity (and sometimes bureaucratic). Be aware that people in politics are very averse to change and are very ideologically stubborn and "the people" is more sometimes more important than logic and facts.
I "rose" quite quickly internally because from the beginning I had a complete different mindset. A doer attitude, new and good ideas, persuasive, know how to speak and most of all, had good connections (and a big car to take people around). Thankfully some of the people followed.
It is important to find like minded people within your political party. Always check water temperature before jumping into the water or you will be promptly ostracized and cast out as an outsider who is trying to change "how stuff is done around here". It's is a bit of a grind.
However you if you have the right attitude and mindset it can also a good thing, because some people might see you as new fresh blood with new ideas and approaches.
Funny story, last year I had an event with very important people from the party, organized by a great guy who is a MEP in the EuroParl.
We had some "debate training" on some topics, and at the end of the training they were asking us if we were for or against on those same topics. One of the topics was "If there should be a EU referendum in Portugal".
In a room of 120 persons aged 16-29 I was the only one saying yes. In front of the MEP and some MPs. Everyone was a supper shocked as this is kind of taboo topic, mostly pushed by leftists in Portugal (they may have thought I was some hidden communist or something). The MEP asked me why, and I replied him quickly with what I believe were very good arguments. He was impressed and we had a small discussion a bit later in the day.
The last day of this event there was a guest from other country, and I made him a very difficult question (in english) and not just some random fluff question done 10000 times. The guest said I was harder than the press in his country and shortly after, the MEP came to me again and said my English was excellent and that the party needed more people like me.
Before speaking, make sure you know your stuff and have your shit straight. Or people will eventually take what you said out of context (call them out on it for not being intellectually honest).
It is also important to know when to shut your mouth. Politics is a game (of snakes and ladders) and by listening you learn, and by talking you give away your position, even within your own party.
Pro-tip: If you fuck up when bringing some topic, or saying anything wrong, just say that you are not closed to any topic and debate, even of bad things, is always healthy.
Pro-tip II: When somebody throws the argument "the best for us all", you can be sure it's the best for that person.
Pro-tip III: Learn and understand body language.
Like work, politics is never ending. There is always something happening will be happening as long as mankind still roams this planet. If you devote to much attention to it, the rest of your life will suffer, be it personal relationships or work. I am giving each time less attention to it, because I am a bit fed up with the beta cucked attittude, the leaflet propagandist mentality (whom is easy manipulated by MSM) and the current political situation in my country (there is no party where i would truly fit). Also being in a relationship does not allow me to do all I wanted.
If you have any question, just ask. I will answer to the best of my abilities.
