There are two main aspects of upgrading the strength of your mech as quickly as possible. The first is how strong the piece of equipment is when upgraded, and the second is the efficiency of fusing any particular item to upgrade that piece of equipment.
Let's start by looking at upgrade efficiency.
Efficiency is determined by a) whether the items share a color type (physical, heat, energy) and b) whether the items share an equipment type (side weapon, core, legs, etc.). There are a few exceptions which I'll get to in a second, but for now, here's basically how it works:
Common items boost: 400, 440 or 480 depend on whether they color/type, neither or both.
Rare items boost: 1760, 1936 or 2112
Epic items boost: 3500, 3850 or 4200
Legendaries and mythicals follow roughly the same percentages.
A few notable exceptions include:
-Triple resistance modules have no color type and therefore can only give 2112 to other triple resistance modules. That makes them good items for boosting rare items to epic, unless you have the legendary version in which case you should use them to upgrade that.
-There seem to be 2 kinds of side weapons, along with several errors in item categorization. For example, side machine guns don't get the "weapon type" boost from many side weapons. There are also errors in categorization. For example, a rare physical sword boosts side machine guns by 2112 (hitting both types) but a rare heat sword gives 1760 (hits neither). These errors and type differences are worth taking into account when you upgrade.
-Extra points don't pass over when upgrading to the next level. So if you had a level 29 epic item that was 400 points from being level 30, you wouldn't want to use a rare item to upgrade it.
So basically, whenever you're boosting something you want the items to match type, color or ideally both. For improving your mech as quickly as possible, it's generally best to get both types matching when you can, but one type is good enough. For example, if you have a "heat" core, you might also use physical and energy cores to boost it, since you can only use 1 core. Or if you're an all-energy mech, you'd use heat and physical items that only match by part rather than color.
To quickly sum up: use items of the same type and color if possible, and always at least one of the two.
Now that you know which items to use for what, which are the best items to upgrade?
You should use these 3 main criteria when deciding:
1) Can the item be upgraded to mythical?
All the best items are mythical items. It makes little sense to upgrade items that can only go to epic or legendary. Do not waste time fusing these items. This is also true of items you don't plan to eventually use -- don't upgrade these items beyond low-level legendary.
2) How effective/impactful is upgrading the item?
Upgrading a Grappling Hook, for example, is not very impactful because it has only one use, doesn't deal that much damage and may not even get used it some matches. Upgrading, say, a core, however, gives many different stats and is useful in all situations.
3) Upgrading efficiency
First, see above. Can the item be upgraded efficiently with the items you're fusing? Also, remember it takes more fusion points the higher level an item is. Taking a mythical item from 49 to 50 takes roughly 25000 points, whereas you might be able to get the same boost in strength by putting 2000 points into another item.
Now, with all that said, here is roughly the order in which to prioritize:
Cores provide many different stats and in relatively large amounts. Whereas a heat module, for example, gives only a moderate amount of heat and cooling when maxed, a core gives significantly more AND energy, regen, HP, and resistances. Even if you have one core in mind that you haven't found yet, it's worth upgrading your current core to epic or legendary just to make your mech stronger, then you can fuse it to make a mythical item later.
These 3 are roughly equal in usefulness, with legs perhaps having a slight advantage. It's generally best simply to apply the fusion parts that are most efficient to each of these parts. So legs would boost legs, weapons boost weapons, and so on.
Generally speaking, modules should be upgraded only when you have the most efficient fusing parts (other modules of the same color) or when you have no other efficient items to fuse two (say, physical mech gets an energy item). While modules are very important, they scale slowly and individually have a smaller impact than big items like your core.
- Teleport, Charge and Grappling hook
These are single use "weapons" that do relatively small amounts of damage (and in teleport's case, often none). These should be upgraded last and primarily with other utility items of the same color for maximum efficiency.
That's about it. Happy grinding