Some of the terms might be confusing you. "A" flagship could be just about any medium-large ship, depending on the context, while "the" flagship is the survey vessel you start random map games with. I don't remember what tech allows you to put survey modules on your ships, but I seem to remember it's not one I usually get that early. In any case, a survey vessel has nothing to do with planetary invasion.
First, before you can invade a planet, you must destroy all enemy ships in orbit around the planet, that is, all ships on the same tile as the planet. Any ship with weapons can do this, but you must be able to overpower whatever ships he's using to defend with. Research Space Weapons and the subsequent technologies to get better weapons. Since (if memory serves me right) the only defense you can get for your ships is armor, and theyuse beam weapons, then armor is irrelivant. What you need to defend against beam weapons are shields, and I don't think you can get those in the first couple campaign missions (the only ones I've played). So focus on weapons.
Second, you need transports to carry troops to the surface to invade. Research (as you said) Planetary Invasion to get access to transport modules. Be aware that, like colony ships and constructors, once a transport ship is used it disappears. That means your transport ships should be separate from your fighting ships. Don't stick guns and transport modules on the same ship, because you will lose that ship when you invade.
Now, on to your other questions. You do not need to invade the Drengins' planets to win. You only need to take them. However you do that is up to you. You can surround them with influence starbases and make them "flip" to you. You might even be able to research a technology victory (not sure about that, though). You only need to win, by whatever means. That will usually mean taking all the Drengins' planets, but there are also a few ways to do that, invasion and infuence, for example.
As per that number, it's as Loupdinour said. That number represents the "size" of the ship. When you form a fleet, the "size" of the entire fleet is limited by your logistics stat. So if you logistics is 10, then you can have three ships of size 3 or two ships of size 5 in a single fleet. I believe you are also limited by your logistics stat in the total number of fleets you can have (not certain of this, I'm more focused on the economy/technology aspect, I don't usually do much fighting until I can build megaawesomsuperduperbombad capital ships i.e. have researched all techs that might enhance my ship designs). The benefit of a fleet is that all ships fight (and move) together. In a very simplified example, let's say you had three ships of strength 2, and was trying to kill one ship of strength 5. That other ship might very well destroy all three of your ships without taking much damage if you send them in one by one. But if you combine them into a fleet, now they have a combined strength of 6 and have a much higher chance of winning. If the enemy ship kills one of yours, the other two will still damage it, and you should win the battle almost for sure. Of course, combat is more complex than this, but it should give you an idea.
Hope that helps.