If you want to be involved: Run for a local office (like maybe a city council seat), preferably an uncontested race where a write-in campaign can be effective with minimal ad spending.
If you want to learn: Join the local chapter for a national campaign. Try to get yourself nominated as a delegate.
If you want to cause a specific impact: Be wealthy and donate significant amounts of money to causes you care about.
As an American in his sixties, my opinion is that the best way to get involved in US politics is to move to Ireland and watch it from afar.
But if you are serious about it, you'll have to learn about what are the major issues your country is facing. Politics is always done on people's problems or issues, they could be social or economics related or both. Everyone have a different perspective too, labor has a different one than tories, democrats have a different one than republicans.
But at the end of the day, remember that power is all that matters for politicians, they never take ideologies or issues as seriously as we plebeians do! For example, in the elections of the Indian state of Maharashtra recently, a political party called Shiv-Sena joined hands with another one called NCP (their ideals are as wide apart as republicans and democrats, or tories and labor!).
So, as a plebeian or citizen, you should call for more transparency and betterment of systems. You should also ensure that you don't hand too much power to one single guy or party because you know, power corrupts!
