How does government ethos change work in Stellaris

Original post:

1. Tech. There are several social techs providing +1 influence/month. Additionally one of thsoe provides an empire unique building with +1 / month.

2. Declare rivalries. You can have three rivalries, together they'll grant 2-3 influence per month.

3. Complete the mandate. If you play a democratic empire (elections every few years), your elected ruler will declare a mandate (e.g. build 4 mining stations), which will grant a significant influence boost (lump sum of 80-200)

4. Reduce expenses. Empire Edicts and frontier outposts cost 1 influence per month. There is also tech to reduce this cost.

5. Some anomalies and events may also grant influence.

Update 2019:

Since release the ways of generating influence significantly changed. The wiki provides a comprehensive list. Additionally, the factions in your empire play a major part in generating influence for you.

Addendum to the old points:

1. Techs are severly reduced. Autonomous agents (for gestalt conciousness) grants +1 influence and living state (for all other races) grants +1 max influence from factions. Interstellar campaigns and galactic campaigns reduce the influence cost of claims. That's all influence related tech left. The empire capital complex building also no longer exists.

2. Rivalries are still there, but only grant 0.5 influence per rivalry.

3. Mandates are still in the game, but now grant Unity, not influence.

4. Empire edicts no longer cost influence per month, but a lump sum to activate for a 10 year period (baseline), outposts are now needed much more frequently than in the release version, they also cost a lump sum of influence based on the distance to your territory and have no influence upkeep.

5. remains unchanged

To win a war you need to get your war score high enough to a point where the enemy will either surrender and agree to the war goals you specified or you can negotiate depending on high your war score.

To gain war score you can do the following:

• Invade a planet - You can bombard a planet, which as you described is what you did but after you've worn down the enemy army and defenses you need to deploy your land armies to invade the planet it conquer it. To do this you recruit the armies, deploy them into space and land them on the target planet. Be warned there is a large amount of fortifications on most planets so it's best to bombard first. If you conquer a planet, this is usually the planet that will be given away during a war/negotiations.
• Win a fleet battle - By engaging in a battle with an enemy fleet and winning you gain war score. I believe this is calculated depending on the size of fleet you destroy, as noted by @Raven Dreamer fighting and even destroying enemy ships does not generate warscore if either you or the enemy retreats.

These two methods will increase war score, however it would be a good strategy to destroy space stations and blockade planets to prevent valuable resources being given to your enemy, when blockading it will stop any tile resources from reaching the enemy without the need to conquer the planet.
Wiki with information