Colonies are very important in Mount and Blade 2: Bannerlord. There are times during the main campaign that you will need to have your own settlements, so knowing how to get them is important.
The easiest way to get a settlement, at least in comparison, is to take it. Find the one you like, then start taking it from who owns it. Find the colony you want to take over, then attack a neighboring village belonging to that kingdom. Navigate to the village on the map, then select "Take Hostile Action". This will allow you to raid the village and will mean that you are now officially at war with the faction that owns the village and the nearby castle that you have your eye on.
When it comes to choosing which castle you want to take over, you need to inspect it carefully to make sure you can win the battle. You'll need to take the full garrison of the colony, so look for a nice weakling to take over. You will need to craft siege weapons to clear the castle walls, as this will dramatically reduce the duration of the battle. Either way, it's a risky business, so make sure you have a good, strong army before you try.
The other way to get a settlement is longer but more secure. You can join a new kingdom and do quests and tasks for them, fight their enemies, and be generally useful. Eventually, you will earn a settlement as a reward.
To join a kingdom, you will need to speak to a lord in a castle. They will offer you contracts, and eventually you can officially join the kingdom.
Either way, it involves a lot of work and risk, and there will always be a serious impact on your diplomacy with someone, so know exactly who you are going to make worse with each path.
