Chilzen gave a really good summary of how to get started. As you proceed further, below is some target gear you'll be interested in from Adoulin. I give a lot of info at the risk of going a little too far, but this should keep you busy a while ;)
Alluvion Skirmish as soon as you can! The Yorcia skirmish light DD set (Taeon) is pretty much the best gear for PUP right now. Even if full augments are too pricey for you now, "Leaf" stones are rather cheap and Leafslit can give Triple Attack +1% to +2%. TA Taeon is THE best TP gear for PUP, period, and is near the top for WS. Can also augment to make more WS-focused pieces (best in slot), or pet-focused gear with -orb stones.
Alluvion Skirmish H2H weapon (Ohrmazd) can be the best non-Mythic weapon in the game, but requires strong augments that can be very expensive to get. A safer bet is to start with Tinhaspa or Oatixur (see below), and once you're more established and have some gil to toss at fate, try for some Ohrmazd augments. DMG+ is the big one, you'll want DMG+20 or more from Duskslit stones, which can be hard to get.
One note is that if you have friends who do Skirmish, you can operate the book in Yorcia without fighting anything - so gear doesn't matter if you learn to use the book. Might want to read up on tips elsewhere for Alluvion Skirmish Yorcia, but basically you're just creating units sort of like an RTS game while your teammates kill stuff to get points that you spend to make units.
Other post-Adoulin stuff you'll want a lot as PUP:
Wildskeeper Reive gear:
* Uk'uxkaj Cap: Drops from Hurkan in Marjami Ravine. Fantastic WS piece, especially if augmented with additional STR (STR augment requires KI from Morimar WKR). Still solid TP piece too, even if not the best.
* Buquwik Cape: Drops from Achuka in Morimar. Fantastic WS piece. Dirt cheap on the AH, but you probably want to do Morimar WKR anyway to get key items to augment stuff so you may get the drop.
Delve 1.0 gear (Ceizak, Foret, and Morimar zones):
Note: can either get via direct drops from the Delve NMs, or buy them with Plasm (Delve currency) once you've defeated any Delve NM of the appropriate tier (NM1-3 is tier 1, NM4-5 is tier 2, boss has boss drops) in the appropriate zone.
* Bladeborn/Steelflash earring set (DA+7% is the set bonus when wearing both, and is PUP's best TP ear set). Bladeborn from Morimar NM1-3 KI (or drops directly from Raptor), Steelflash from Ceizak NM1-3 KI (or drops directly from Scorpion). Get these right away, they're fairly easily obtained and even if you don't get a drop they cost very little plasm.
* Asperity Necklace: Good for most jobs that can use it, though non-Mythic PUP can almost always get by with Cirque Necklace (which is often still best in slot). Asperity comes from Foret Delve NM4-5 KI (or drops directly from Nerrivik)
* Oatixur (H2H) and Hurch'lan Sash: Both drop from Tojil, the boss of Morimar Delve. Oats are not as good as some weapons like an upgraded Tinhaspa (Delve 2.0 zones drop) or a well augmented Ohrmazd (Alluvion Skirmish weapon), but they're still respectable and near the top of the heap - and probably little competition these days for the drop. Hurch'lan Sash is a nice Master+Pet belt with good haste and accuracy for both.
* Whirlpool Mask (Dakuwaqa, Foret boss): Still very solid Acc/Atk headpiece. I dunno that I'd spend plasm on it if you don't have much, but a great drop item that's still competitive with top options.
Delve 2.0 gear (Marjami, Yorcia, Kamihr zones):
Note: these are pretty difficult events, but at least you should be able to get KIs for the non-boss tiers (NM1-3 and NM4-5) with a little help, even killing the NMs outside of Delve provides access to the KI.
* Tinhaspa (H2H): From Kamihr NM4-5 KI, or drops directly from Mirka. This is the standard high end weapon for PUPs these days, beaten only by Mythic, Empyrean, or a very well augmented Alluvion Skirmish H2H. You'll want to augment them to rank 15 on the "B" path (Accuracy path), and fully augmenting will cost 6 Airlixir+2s (or a lot of NQ/+1 Airlixir). It even has automaton Atk/RAtk/MAB stats!
* Ighwa Trousers (Kamihr mega-boss): There are other options, but these are still an excellent WS piece for Stringing Pummel.
* Qaaxo gear (various): Can be augmented to be good endgame stuff, though it's now mostly beaten by Taeon so getting more irrelevant. Body in particular is still quite nice.
* Chaac Belt (Marjami mega-boss): Treasure Hunter+1 for any job. With this Belt (macro it in for a single hit) and /THF, PUP/THF can get TH3 - a really nice job for farming Dyna or Salvage for currency/alexandrite.
JSE Cape:
* Dispersal Mantle: JSE capes come final rewards from Reives (WKR, Colonization, Lair) or from the Incursion event. JSE capes are random between all 22 jobs, but you can trade any 3 for 1 of your choice at the Library in Eastern Adoulin. PUP cape is a nice Acc piece with some good pet stats. Can augment it further with random augments for the cost of a Refractive Stone (AH or buy with 15,000 Unity points) - for PUP, Martial Arts is the really good augment you want, but can also get STR, DEX, and automaton TP bonus. Incursion drops come pre-augmented. You can overwrite augments.
Skirmish 1.0 armor:
Otronif set. Original pieces drop in Yorcia Weald Skirmish (Skirmish 1, or, NOT the new Alluvion Skirmish event in the same zone). They're upgraded to item level 119 with higher stat vomit by getting key items in Outer Ra'kaznar Skirmish.
Otronif remain useful for DT- sets, as PDT- augments are common and go up to PDT-4% (best in slot for PUP in most slots). If you augment the NQ item level 115 version, augments are wiped when upgrading the gear to +1 ilevel 119 version, so take care. I'd recommend trying to get to 119 first, since if you're making a set focused on DT- the substantially higher defense of the 119 pieces is definitely relevant.
You can also use "wings" items that drop in skirmish zones to buy armors, 1-3 Lebondopt Wings (Yorcia) per NQ piece. 6 Pulchidropt Wings (Ra'kaznar) to buy the KI to upgrade a piece to +1. Note that Leb Wings sometimes show up in login campaigns (including this month) and Mog Pell rewards.
Don't bother with non-Alluvion weapons (Ninzas are the H2H), they're not worth it.
Reforged AF/Relic/Empyrean Armor:
Luckily or unluckily for PUP, depending on your perspective, most of the item level 119 reforged gear for the job isn't that amazing so not a necessity to upgrade.
AF: None of the pieces are particularly good, don't stress about any of the set.
Relic: Body is still quite nice as a master and pet hybrid piece (or even just a DD pet focused piece). Head is a good idle regen and automaton refresh/regen piece.
Empy: None of the pieces are BAD, it's just that Taeon is better gear and is easier/cheaper to get than upgrading Empy to 119. Karagoz legs are probably the best of the bunch, but I'd focus efforts more on getting Skirmish armor.
Job/Gift point #100 changes everything. so get there ASAP.
No, grinding 100 job points does NOT change everything. Two extra elemental slots for automaton attachments is certainly a nice bonus, but it doesn't fundamentally change the way your play the job or make you THAT much more effective. Just allows for a slightly more optimized automaton loadout.
If you have to choose between spending your time getting gear in other Adoulin activities or grinding conquest points/JPs, definitely do the other stuff.
Also note that PUP's job point categories are
really disappointing. As in, maybe the worst set of JPs of any job in the game. They're bad. The gifts that come along with a certain # of JPs spent are helpful, but not game breaking.