the gear from the Peacekeeper's Guild is likely a step up from sparks gear (save for the weapons, those are actually solid until you start getting "real" 119 items) if you are keeping up with Tutorial Records of Eminence quests, they will toss you a bunch of REMs tales which will give you a quick head start on upgrading AF/Relic. additionally, you can get some various skirmish pieces which can be really good with a minor gil expenditure. otherwise, if anyone can let you leech some Reisenjima or even Zi'Tah gear, that will put you on very solid footing.
upgrading your AF can be important (especially in the face of Omen) or it can be a foolish expenditure, and this is going to vary from job to job. Dragoon AF is laughable, for example, in the face of most other gear (arguably, AF Legs is the only piece worth upgrading in Omen). I don't know what job(s) you're settling into so I cannot offer input regarding that, but refer to Forum guides here, or on BG for some insight. additionally, your accessory slots are going to be sorely wanting because so many of those are dependent on group content; even the basics tend to be Delve items or better. it's still worth getting a Brutal Earring, Rajas Ring, and Epona's Ring as they have aged surprisingly well depending on the jobs you are looking to play. further, completion of Mission battlefields is required to do the high level versions which people are going to be spamming hard come next week, and offer many pieces that are relevant for someone looking to catch up. additionally, get whatever Ambuscade gear you can. this is pretty much the best "free" gear available to you, and the Job Specific Capes are best in slot for almost every situation you can tailor them to.
as has otherwise been stated, definitely look into leveling some sort of support role to get yourself into the higher tier of content. DD spots are going to be highly fought over and are going to be given to those with the best gear. there is an XP/CP event beginning in a couple days, you should try to make use of that while you can as well, as capping jobs is tedious af, even when you have every RoE boost to CP gain.
now that I have given a daunting wall of text, I will end with this: try and mix and match activities. keep your activities varied to avoid any sort of burn out - because the game has been around so long there is SO MUCH to do.