They all do different things:
- Alchemy Lab is a source for emblems and unfarmable mats See my guide for more words on the subject
- I pretty much only use Hunter’s Lodge to buy harpoons for the defense down effect on Titans. The rest of it is trash unless you’re hunting 13 and 14-star Titans with your alliance
- Hero Academy is, like the old version of the Alchemy Lab, an expensive, wasteful boondoggle. Consider building up to Academy 10 so you can trade in Legendaries, but realistically only expect it to pay off maybe two or three times per year, and don’t even bother researching any of the other levels