Make sure you've read the in-game tutorial (question mark button at the bottom of the screen). Most of this guide just reinforces material from there.

In a standard run, your goal is to govern for 25 years and get a high score. You should learn to play a standard run before playing campaign.
At the bottom of the screen, there is a button for Statistics that has a tab Score, which shows how scoring is done. Do check it out, this guide doesn't cover it in full. Most importantly, you get a lot of score for Planet Levels while Happiness is a global score multiplier (meaning it affects everything)

The most important concept is upgrading as many planets as possible to the next level.
Click on any planet and pay attention to the text in bottom left to find out how to do this (all planets are upgraded roughly the same)

A Struggling planet is much worse than having nothing at all! If it never upgrades to Established, it can end up like this:

This one planet is not just bad, it reduces your whole final score by 10%. Make sure every Struggling planet can upgrade in a few years or else the negative happiness will remain permanent, even after upgrade:


Established planets are neither bad nor very good. A sector that has only Established planets is mediocre. Make sure your planets are not just trading and surviving but actively growing or helping other planets grow to Successful and Prosperous, which are the only levels that are a benefit to your happiness, which increases score.
(Successful)

Undo is extremely important. If you got a Struggling planet and can't upgrade it soon, you should undo.
A few actions that reveal new information prevent undo: Sending Probes, looking at Council Tasks and activating Forebear Ruins. In the campaign, some actions also reveal information; usually denoted by a yellow exclamation icon or description text informing about new knowledge.
Try not to send any probes until all your planets are Established or better. If you think you will reveal a nearby planet that gets you what you need, it's often better to undo and not rely on further scanning.
Go wild! Colonizing planets and connecting them up can do no harm as long as you can undo. The best plan is an executed plan and if it didn't work, undo and try again.

• Science is very important if you wish to have a high score. Note: technologies aren't worth anything by themselves - their purpose is to help with your primary goals, usually upgrading planets.
  
• Forebear Ruins are always a positive. Make sure you explore as many as possible and get their rewards (reminder: you cannot undo if you explore them! be very careful)
  
• Tasks and Council Levels are a direct secondary source of score. Try to pick easier tasks that you can complete as soon as possible instead of looking for bigger rewards. At the end of the game, unfinished tasks are counted negative! Do not take any tasks you cannot complete. In the last 5 years, you can take fewer than 2 tasks and you should use this; don't over-promise. (reminder: you cannot undo if you check newly given tasks, careful!)
  
• Money is merely a tool and does not contribute to your score - at the end of the game, it's worthless! Most technologies and choices that increase money are only useful in the highest difficulties where it's very hard to come by. If you have good planets, money will not be an issue and you may often end up having a surplus. A good sink for money is the Lab - its maintenance costs quite a bit so make sure it doesn't cripple your economy but as soon as you have any surplus, it's a great use for money.
  Inflation is simulated like so: in the top left, your empire size and, later on, slipway overload are shown, which increase planet upkeep and slipway cost. It does not increase if you don't expand. For a secure income, try not to settle too many planets and anticipate a massive fall after you reach the next tier of empire size (but don't hoard money for this reason.)
  
• Extra resources are useless! In fact, if you don't export any resource from a planet, it becomes unhappy. Resources are only any good if they have a destination (reminder: a planet will always be glad to accept more than it needs; use this!) Labs are better placed strategically with access to many resources but, in a pinch, can act as a dump for export, both of materials and of people (who are very happy to work at even empty labs, increasing score!)
  
• Time is the most important finite resource. You have exactly 300 months until the end of the game and the more efficiently you use these months, the better the score. Slipways take 1 month to build and Colonization takes 3 - these two actions will take the most of your time, by far. Try and find ways to achieve your goals by settling as few planets as possible and that way you will be more efficient in terms of time. The one big thing to avoid is the button "skip to next year" - you should avoid getting into a situation where you need to press it, it is the worst use of time.

To start off a run, try to have a self-sustaining loop. In fact, it's always guaranteed there is one in the initial circle.
There are a lot of helpful tools in-game, including settings for sticky keys and easier probing, hotkeys for some actions and blueprints/ghost plans for future colonies (temporary by holding Alt and persistent by right-clicking a colony type to "pin" it).