It's funny to hear that it's so many to keep track of, etc. Not that it isn't a fair bit, but the hull-type-based bonuses were added like a year and a half after the initial release of the game. They replaced
ship-type-based bonuses
At least now you know that if it's good against one ship type (bombers) with polycrystal, it's good against other ship types with polycrystal (like space tanks). At initial release there was a lot less going on in other parts of the game so the added detail was nice, but I digress.
Anyway, if you'd like to know more about what a ship is good against:
1) As Commiesalami said, check the hull-type-based bonuses in the normal tooltip for the ship.
2) On the Stats window (opened by the "STATS" button in the lower-left of the screen) on the Reference tab, you can get a comparison between any ship type and any other ship type. If you're into excel nerdery (I certainly am), you can even export a spreadsheet with the various balance stats, including hull type and DPS vs each, etc. Not that the spreadsheet is necessary, of course
3) And a relatively recently added feature for getting at the info:
** Added new PlanetView KeyBind: "Show Strong/Weak Info":
*** Defaults to Alt+W.
*** When this is active and the mouse cursor is over a ship, each planetary summary sidebar entry will display Win/Lose/Draw (and a % indicating intensity of win or loss, if it's not a draw) as a rough indicator of how effective a cap of the ship type under the cursor would be against a cap of the entry's ship type. Since planetary summary entries frequently include multiple distinct types of ships, this is perhaps most useful in conjunction with the "Guide" mode of the planetary summary that shows all ships of a given mark level (the default key for switching to guide mode is F1). However, even in normal mode this can be useful to at least get a rough feel for which of your ships on the planet are the best you have on hand against a specific target.
*** Caveat: this is not using a full simulation or anything like that, it uses the same simplified formula as the reference tab uses. So stuff with modules won't really get accurate results (because it's only counting the base hull), etc.
*** Suggestions on text color, default keys, etc, are quite welcome; we're not totally happy with the effect right now but don't have a lot of time to fiddle with alternatives.
** Added new PlanetView KeyBind: "Strong/Weak => One Vs One":
*** Defaults to Alt+E (so the full combination would be to hold Alt+W+E).
*** When both this and the "Show Strong/Weak Info" keybind are active, the strong/weak info shown will use a one-ship vs. one-ship comparison instead of the normal cap vs. cap comparison.
So you can put your mouse cursor over one of your ships and hit Alt+W and it will display a comparison on each icon in the sidebar on the right.