In general, I'd say that this is the order you want to level up in:
1. Dual skill SRs* (we have very few good ones yet)
2. Dual skill SSRs
3. Relic weapons
4. Single skill SRs*
5. Single skill SSRs

* This only applies until you can get a full Grid in a reasonable timeframe. For example, in the next two months we'll have one Thunder event which gets you two Assault SSRs. So only keep leveling SRs that'll stay in your Grid even after this event. If you look further, there's two more Thunder SSRs in late March.

Note that this only applies to Assault weapons. There's no point in leveling up a Defender/Exceed SR for example.

The reason why I recommend SRs first is because each skill level gets you the same amount of gain. And each skill level is literally twice cheaper in SRs than SSRs.
Quote Originally Posted by toastedsnow View Post
--
Not sure how helpful the picture is so here's a list - thunder weapons only - as well:
Dazzling Gold Carbine +1 (SSR)-(LB0) - Skill level 5
Lightning Drum Hammer +1 (SSR)-(LB0) - Skill level 5
Charles' Thunder Blaster (SSR)-(LB0) - Skill level 1
Side-by-Side +4 (SR)-(Fully LB) - Skill level 4
Side-by-Side (SR)-(LB1) - Skill level 4
Lightning Strike Axe +2 (SR)-(Fully LB) - Skill level 4
Lightning Strike Axe (SR)-(Fully LB) - Skill level 4
--
There's two things that needs to be said about this:
Hime only gain skill level power-ups from weapons that match their element. So those two Light Axes and that one Fire Hammer only give stats to Thunder Hime. That's fine for a new player, but eventually you'll want to replace those with stuff on-element.

Secondly, you have a Relic weapon (Charles' Thunder Blaster) in your Grid. Those weapons ONLY work in the main-slot and ONLY for the specified Soul. In other words, at the moment it does nothing for you. It needs to be equipped by Dartagnan, or it'll be considered off-element again.