Research the Logistics tech tree branch (starting with Basic Logistics).  If doing such research doesn't help you make bigger fleets it's possibly because you're playing one of the campaigns.  Play sandbox (new game) instead.

You can also later on build the Hyperion Logistics Centre which gains you an additional 6 logistics points per fleet.

MarvinKosh is correct, but I might add a couple bits.

First, some races may start with larger (racial bonus) logistics value.

Second, though one may build only ONE Logistics Center, one may CAPTURE others and gain +6 for each.

The above two factors is why you will sometimes discover that an enemy AI has a larger fleet than you can assemble, despite the fact that you have maxed out the logistics research tree and built the bonus building.

This is the same effect one gets by capturing extra Hyperion Shrinkers, in that one gets the additional space gain for each additionalvshrinker captured.

In Suicidal games, it is sometimes important to ID the enemy shrinkers, logistics buildings, soldier bonus, etc structures and target those worlds for early capture.  Depriving an enemy of the bonuses while adding them to yours is often a turning point of sorts.  In some really tight Suicidal games, I knew I could not hold the planet after capture, but it takes only one turn to demolish the structures or (if you are really hard core!) turn the planet into a permanent PQ 0 wasteland.

You can attack a ship in orbit around a planet with any ship that has an attack factor.

You cannot attack a world.  You must invade it.

To invade, you must research Planetary Invasion and then build ships with troop transport module(s) on them.  Next, launch them and (just as in Colonizers) you get asked how many troops to load.  Then you click on the invasion target just as if you were trying to attack the planet or land on one of yours.

One cannot invade a planet with a ship still in orbit, so you must destroy any orbiting defenders forst.

When you invade, you will get asked / offered various tactical choices.  There are several factors that are considered, including invasion/soldiering tech dependent ones.  Look over the invasion and planetary research lines.

You also can gain planets without even being in a war.  One way is by Influence flipping, another is by simple trading/buying, and a third is when one Empire surrenders to another.  In the third case, an ally will generally give you its remaioning planets if it surrenders to an enemy.  You may also get them when a non-ally surrenders to an enemy, but you have one or two treaties with the losing empire.  You may also get them in other similar cases, such as being "Close" but not an ally, but this is less predictable.