Dominion is a much different game mode compared to Summoner's Rift. Rather than just trying to lol power leveling kill your opponents and destroy their Nexus, you have to capture and defend control points until the other team's Nexus eventually runs out of health.
If the enemy controls the top capture point, and has several champions on it defending, then go capture their mid point instead of trying to overtake them.
Turrets on Dominion can make a huge difference in battles, sometimes even inflicting thousands in damage. You will be very hard-pressed to take a point when there are three or four enemies on it defending. A good rule of thumb is to count an enemy turret like another champion. So for example, if there are three opponents standing on one of their turrets to guard it, count it as if there were four opponents there. Even if you have four allied champions try to take it, you don't really have the advantage despite the fact you technically outnumber them four to three. If you instead ignore that point and go take their mid, this will force them to stop defending and either come and stop you or try to take your mid. This is called "backcapping."
Either way, this is a good thing because it takes away their advantage by forcing them from their defensive position. If they try to come stop you, you should be able to neutralize their mid point before they get there and then you will have a battle on more even ground (as opposed to fighting under a turret they control). If they go to capture your mid, you can nullify this by capturing their mid and then going up and capturing top as well.
Ambushing doesn't help you
One of the biggest things that people need to watch out for on Summoner's Rift is being ambushed (called a gank). Summoner's Rift players tend to want to carry this strategy over to Dominion by waiting in the bushes in the middle of the map for people to walk by or ambushing them when they are trying to capture the Storm Shield Relic in the middle. Even if you kill them, this doesn't help you because respawn timers are much shorter on this mode. Dominion is not about getting kills – it's about getting captures. The time you spend waiting for someone and then ambushing them is time wasted that you could have spent attempting to capture another point or defend one of your own. Although getting a kill may feel satisfying, waiting around in the bushes only hurts your team in the long run.
This tip goes hand in hand with the previous one. This game mode isn't about kills so don't chase people around the map. Again, it's only wasting your time. Delaying your opponents is a big part of Dominion so you are playing right into their hands by chasing them away from a point that you haven't captured yet. The capture rate increases with more champions capturing a point so even if you have an ally capturing it already, there is no excuse to chase. You'll still be capturing slower than if you didn't and that time could mean the difference between a win and a loss in a close game.
Another tip worth mentioning is not to waste your ult ability on a fleeing, low health opponent. It doesn't really matter if they get away and go back to heal. Wasting your ult on them means it might not be available to you in an important team fight when you need to capture/defend a control point. Use your ult in situations where it will actually help you do so.
As long as you control at least one capture point more than the enemy team, their Nexus's health will diminish as time passes (generally this means you control three out of the five). One of the biggest mistakes new players to Dominion make is that after they capture a point, they immediately want to move on and try to capture another one. By the time they get to it, the enemies have respawned and kill them. Then they simply lose control of the first point they captured! Playing defense is just as important as actually getting the capture in the first place. If a majority of your opponents are dead, you might be able to neutralize one of their capture points before they respawn but keep and eye on their respawn timers. You shouldn't be pushing for another control point unless you outnumber them by at least two.
Don't stand right on the point when guarding it
When you do stay at one of your control points in order to guard it, don't stand right on top of it. The enemies can see you! If they know where you are at all times they will be able to plan for ways to beat you – whether this means backcapping a point they know is unguarded or knowing exactly how many people to bring to kill you.
Instead, stand near the health pack that spawns next to your control point. This area of the map is still shrouded for your enemies so they won't be able to see you. They might not even know you are there at all! There are some bushes next to each one that makes a perfect waiting spot. This is an exception to the "ambushing doesn't help you" rule because you are actually defending one of your control points and not just ambushing someone for the sake of a kill.
Have a little patience. Wait for an opponent to start capturing your point before popping out and letting them have it. They will be defenseless for a brief moment when they start the capture animation. This gives you a second to hit them with an ability before they can react. Make the most of it by opening with a stun, taunt, or snare if you have one.
Here's a fun little tactic you can use to throw off some opponents: if you are hurt after a battle but need to stay on your point to defend it, hit the recall button as if you are going to go back and heal. Then stop it, as if realizing that you are leaving yourself out in the open while recalling, and instead walk to the health pack. If your opponents are paying attention to their minimap, they will see the circle around you indicating you were recalling. They might think that you went into the bushes to recall instead as that is standard practice for many players. Thinking you left the point completely undefended, they will be taken by surprise when you pop out to defend it.