I can see there are multiple objectives I could try to accomplish, but what do they do? Is there some benefit to accomplishing them or penalty for failing?
Completing the objectives will benefit you greatly. For example, destroying the data centers will decrease the AI progress level, and capturing the advanced research stations will net you a new ship type. There's no penalty for not completing them.
Also, is there any way to know how large a force is going to come through a warp gate on a wave?
Are you playing version 1.013? If so, then there's no easy way of doing so. Just be aware that the number of ships will scale with the AI progress level, and the individual wave modifier, for example 2.00X, means the wave will be twice as large as the base size for the progress level. The number of ships is also scaled with the ship cap for that ship type, so you will be attacked by far more laser gatlings than you would fighters for a given wave strength.
In version 1.014 you're told precisely how many ships are on their way.
Ships have all sorts of +/- against other ships, but I haven't found a way to really make use of that. When I'm invading a planet, there's usually several different ship types and they're all intermixed. I've just brought a mix of forces and point and click to kill groups at each command post. The only time I really make an effort with ship types is against certain fortifications (e.g. cruisers against spec ops due to the range). Is there some secret I missing to making use of the ship specific strengths/weaknesses?
You've pretty much got the right idea there. The relative ship strengths and weaknesses ensure that you need to bring a good mix of ships with you in order to ensure that you'll be equipped to deal with whatever the AI throws at you. Manual targeting should rarely, if ever, be required - ships will generally do a good job of picking targets that they're effective against.
Of course, if you want to destroy a specific ship type, say a cruiser for example, then you'll know that using fighters would be much more effective than bombers.