Yeah, you should be making a good amount of Mendacity just from first time bonuses you'll get from catching up on dungeons, on top of running Rabanaster/Ridoranna a few times (you'll want to be saving up those coins for when the new Ivalice raid comes out in a few weeks). With that, you should be able to start grabbing a healthy supply of the i360 tome gear on top of any dungeon drops. I definitely recommend catching up with the MSQ to get all the story dungeons, and also don't forget to unlock Fractal Continuum (Hard) and Saint Mocianne's Arboretum (Hard), particularly the latter to unlock the Expert Roulette, which is very helpful for farming Mendacity and especially Genesis tomes. Genesis gear can get you i390 gear, which is very nearly as high as you can get atm without doing Savage fights. There's also dungeons and trials to unlock through the Four Lords sidequests, so you should definitely look into that as well. Beyond that, it shouldn't take you long to be able to do Sigmascape and eventually Alphascape, the latter of which will drop tokens that can be collected to trade in for decent i380 gear. If you're a glutton for punishment, you could also consider catching up on Eureka for the relic gear, but this likely won't really pay off until you finish with Pyros,which could take some time (you do get some tomes from Notorious Monsters too, as a note). If you like doing treasure maps and hunts you can get a few tomes for doing these on the side as a supplement as well. The Hunt seals can get you the i330 gear, but at this point you'll be finding much better gear pretty fast, so I'm not sure I'd bother with it unless you like the glamour. Don't forget, Genesis tomes are capped off at 450 a week, and you can only hold a max of 2000 of any kind of tome, so spend your tomes wisely! If you fill up on Mendacity, spend those tomes immediately so you can save up some more as Mendacity has no weekly cap. This may all seem overwhelming at first, but once you start getting momentum with your first clears I think you'll be surprised how fast you catch up. Best of luck!
Edit: oh yeah, as for weapons, you have a couple options. You can choose to try your hand at EX trials to farm their weapons. Susano will give you a humble i320 weapon, Byakko will give you an i355, the EX version of the 4.3 story trial will drop i365 pieces, and finally Suzaku will drop a hefty i385 weapon! They'll be tougher fights if you're not used to them, but can be worth it as they're not as bad as Savage fights and are certainly doable for even casual players. Aside from that, if you don't want to try the EX fights you may want to consider using hunt seals for the i330 weapon going into Sigmascape 4.0 normal (O8N), as you'll need to beat that one at least four times for crystal tokens to trade in along with 1000 Mendacity for the i360 weapon. This same process follows for Alphascape and Genesis tomes, though you can currently only get one crystal a week from O12N, and you'll need seven for the i390 weapon. Also, if you really farm the Ivalice raids a lot you can use one of each coin to get items to upgrade the i360 tome gear into i370 Augmented gear!