Eligma 1 - try not to waste your Eligma on units that can easily be farmed in the raid/JFD shops or hard difficulty stage students or students that have niche usefulness. However, exceptions apply, you may do it for hard difficulty farmable students if you really quickly need to boost them up to 4* or 5* and even then I would only do it for the really strong DPSers - Iori, Aru and Haruna. Even then, do it sparingly. Same goes for Koharu and Azusa as it can take quite a while to get enough T2 GA currency but don't overdo it. NEVER do it for any other raid farmable students, especially not Eimi, Tsurugi and Sumire as you get a ton of JTF coins and they're the only 3 students to spend it on.
Eligma 2 - Scam. Avoid. Eligma is way too scarce to waste it on even the admittedly good things that the shop provides.
Raid/JFD shops - get all the girls' Elephs that are on offer and exchange them when you get to 120 of their Elephs, then if needed get them to 3* (like Kotama, Momoi, Hanako). You may see somewhere that Eleph exchange is a bad deal, and yeah it is, if you roll a duplicate after claiming a student you effectively waste 90 Elephs, but raid coins are plentiful and that advice is like 3 years outdated, there's way too many units now for that to be an effective use of your Eleph. After you do what I said, you're pretty much free to spend it on whatever - get the mats, more Elephs for your favorites to get them to 4* and beyond, whatever you want. For JFD after you get Eimi, Tsurugi and Sumire, if their kits don't fit into your playstyle you can immediately shift onto getting the crafting fusion keystones. These are a godsend later on, as they allow you to do things like transform skill books/blu-rays of schools that have low competition for these items due to the small number of playable students (so Arius, Red Winter, Valkyrie) and turn them into upgrade items for schools with shitton of competition for them (Trinity, Gehenna, Millennium).
PvP shop - use it to charge your AP, if you have 50 coins left over after a single charge (so 95 coins at the beginning of the day minimum) you may spend it on the Elephs of a girl on offer, but I only recommend doing that if you like her. Thus you may consider the students whose Elephs are in the PvP shop effectively unfarmable. The rest of the shop is a scam.
Expert shop - believe it or not, but the best way to spend them early on is on exchanging them for 5 mil. credits. Then maybe the pink reports. Once you hit level 90 you stop gaining XP and start gaining Expert permits instead, you want to prioritize the credits, then the limited elephs, then the firing pins, then the rest. However after a few months of being at level 90 you'll get so many fucking Expert Permits that you can literally buy out the entire shop before it resets and still have a bunch left over. You should expect to hit lvl 90 after half a year of playing or so.