I'm still experimenting with the new formatting possibilities of this platform. So if you find any of the changes particularly awesome or abhorrent, please tell me.
Recommendations:
Mods
Primed Continuity
+55% Ability Duration. I cannot emphasize enough how fantastic this mod is. Duration is super important for many builds, especially for stuff that also needs range. This mod used to be The No.1 must have. Nowadays, it has an alternative in the form of Archon Continuity - but that mod's modest and very situational bonus effect is offset by its higher drain and difficult acquisition. I still consider the Primed one as the preferable option and a very much worthy purchase.
Primed Smite Corpus
+55% Damage vs. Corpus for Melee. These anti-faction mods provide respectable boosts, especially since status effects double dip on them. But in normal play, they require too much micromanagement to be effective. There are a few endgame activities that are both exclusively Corpus-themed and warrant a specialized build anyway, but since melee is a poor matchup against Sisters and the Contagion Zaws for Profit-taker don't need more damage, the only usecases for this mod specifically is long-haul Index and Steel Path endurance runs. If you are frequently playing these activities, and have the budget to max out the mods, consider buying these.
Tempo Royale
A nice, fluid stance for Heavy Blades, with lot of spinning and ground slamming. It is farmable in-game, but the droprates are terrible. It's worth the ducats you'd spend on it, but don't prioritize it over other things: it's rival stance Cleaving Whirlwind is a perfectly functional sidegrade, and much easier to farm.
Peculiar Audience
Upon killing an enemy, 60% chance Skittergirl's laughter will play. No tangible gameplay benefit. A waste of modslots, if you ask me, especially since it goes on your Warframe. But cosmetics are cosmetics...
Weapons
Baro's weapons typically fall into low A or B rank in tierlists, so DE at least tries to justify their asking price. And even the trashiest of weapons are worth buying if you need the Mastery Rank they provide. Or if you have good riven for them, since their disposition is usually quite high. On the other hand, Warframe has a massive arsenal, and for every weapon Baro sells, there's gonna be another that can do the same job just as well, or even better. In that sense, everything in here is "safe to skip."
Prova Vandal
Even if it doesn't look like it, this is a machete-type weapon. Unless you really need innate electric damage, this is slight downgrade from the Machete Wraith - which still makes it rank high in its class. But as machetes are overall a middling weapon type, you should probably just wait until the Machete Wraith comes around and get a better deal.
Prisma Dual Cleavers
A dual sword with high stats except for its middling base damage. Also has a mediocre syndicate augment. It's a worthy purchase on its own, but its weapon class is crowded with good options that can make the Prisma Dual Cleavers somewhat redundant.
Cosmetics
(Not Really) New!
A texture swap for the base Paris model (though it also works with Paris Prime). It was previously only available from limited-time events, but now it found a permanent hope among Baro's offerings.
Not that anyone missed it, though.
Ki'Teer Presence
When you just want to stand there, but IN STYLE. And it only costs credits!
Limbo Immortal Skin
A paintjob that covers Limbo with mostly vertical stripes. Can be used to combine Prime details with the vanilla skin.
Ki'Teer Foros Armor Set
A minimalistic armor set in the signature Ki'teer style.
Gorgon Towsun Skin
Twin Grakatas Towsun Skin
If you want your guns to look like they were handpainted by babushkas, these ones are for you.
Renayla Sugatra
A sleek cord-like charm for melee weapons, in the same style as the Vericres warfan. Originally a Tennocon stream giveaway, now available for everyone.
Ki'Teer Straed Syandana
A Syandana in the new Ki'teer style: big, sharp, angular and full of protrusions. Unfortunately, the lower half has jiggle physics used in a strangely linear manner that makes it seem like the textures are being applied to the wrong skeleton.
Masker's Theodolite Crewsuit Set
Another Leverian prop - this time, an Operator cosmetic. If you want to dress your Starchild as a Crewman, this is the closest thing to it.
Kavat Sentinel Accessory Set
This one makes your Sentinel look like a Kavat. A very, very, very, very inbred Kavat.
Ki'Teer Stencil
A pattern for the walls of your personal quarters. These usually cost plat, so take advantage while you can! Or don't, if you prefer your walls clean.
Mahd Luxxum Ornament
Abstract art for your Orbiter. Not particularly thrilling on their own, but can funtion as pedestals or part of a larger structure.
Deimos Fass Prex
Prex cards depicting Deimos-associated entities. These were given away like candy during the Heart of Deimos update. If you haven't got them then, you can get them now for a modest price.
Noggle Statue - The New War Teshin
Bobblehead of our trusty Dax mentor.
Fae Path Ephemera
One the older step-Ephemeras, and rather insubstantial compared to, say, Eros Wings or Vengeful Charge. A permanent fixture of Baro's repertoire.
I might have sharp and opinionated remarks about these items, but I'm not the fashion police. Buy whatever cosmetic tickles your fancy!
Other
3 Day Mod Drop Chance Booster
The same booster you get from the Sortie or login rewards. They don't stack with them by having their effects multiplied - only by having their duration added. The boost also applies to Endo drops. As this particular booster cannot be bought from the Market, it might still worth spending your ducats on them. Assuming you will be farming for certain mods or Zariman arcanes, or run Rathuum for Endo in the next few days.
If you haven't played Sands of Inaros yet, buy the quest key and do it (and bring your best setup for scanning Kavats). If you have Inaros Prime, you can buy the Baro Void-Signal (one per visit) to access some special cosmetics.
Also, check the answers below for second opinions.
The best part about peculiar mods is forgetting you have one equipped when fighting the Profit Taker, and you see the giant flowers spawning on it. I've heard that Growth works too, but when I tried it out, I didn't notice anything.
Oh and you can't hear Peculiar Audience over the sound of your own gun so it's pretty pointless.