Syndicate weapons are pretty much the best in the game, aside from a few of the primes which are competitive. I think everyone should have: Vaykor Marelok and Hek, Synoid Gammacor and Simulor, Sancti Tigris, and if you use Ivara at all or just like bows, Rakta Ballistica and Cernos. The secondaries require MR6 and the primaries are MR12 though. Never spend your own syndicate standing on weapons. The mods cost 25k to buy and the sell for 10-15p. The weapons cost 100-125k to buy and sell for 20-25p. I chose Steel Meridian as my main, and I maxed out Red Veil using the allied rep. Next I'm going to do Suda and Hexis, which requires only minor upkeep to stay in Red Veil's favor. The other two will always hate me, oh well. Bottom line, choose syndicate based on the mods they have and how well you think they will sell, or even just based on which one has a syandana that you like. I'm a fan of Suda's syandana myself.
Since you said you were still MR7, I would advise getting some non-syndicate primaries such as Tonkor (nothing beats this for defense missions, especially while leveling frames at Eris/Akkad and not being a leech), plain old Hek with syndicate mod, Soma Prime, and again if you like bows you can't go wrong with Dread and Paris Prime. If you do sorties you'll have to pick up a sniper rifle for the restricted missions, Lanka and Vectis Prime are the best choices there. Your Sobek is really useful for low-mid level missions if you get the Acid Shells mod from Kela De Thaym.
For secondaries, definitely try to get the syndicate weapons I mentioned, especially Vaykor Marelok. Some people love Lex Prime and say it's the second best, but I think it's fairly shitty myself. Your Atomos and Acrid are still really good when modded properly. Sonicor is one of my personal favorites too, mod it for blast and blowing shit into orbit never gets old. Notable primes besides Lex are Akstiletto Prime, Spira Prime, Hikou Prime (just vaulted unfortunately). And for xp farming at Akkad, Kulstar is a must-have for when you want to level a primary and thus can't bring Tonkor. Beware though, unlike Tonkor this thing will kill your ass. I also think everyone should build a Furis and use the syndicate mod with it; the proc is extremely useful.
Melee is a lot more simple, all you really need is Galatine Prime and Redeemer. Jat Kittag can be fun to use too with the Kela De Thaym mod, and if you want something for Excalibur just pick one of his preferred weapon types with the cheapest stance which is generally a nikana or dual swords. Use your Orthos until you get Galatine P, or just fill the slot with mastery fodder since it's MR13 for Galatine and you're going to need a ton of rank XP to get there.
Re: frame stuff -- augs come from syndicates, most aren't really useful except in very specific circumstances. The must-haves are things like Chilling Globe for Frost, Despoil for Nekros, Resonating Quake for Banshee, Infiltrate for Ivara, Hushed Invis and Irradiating Disarm for Loki, Vampire Leech for Trinity, Pilfering Swarm for Hydroid, Capacitance for Volt... maybe a few others I've forgotten.
Focus requires lenses which come from sorties and other players (lesser only), and the market (greater only, 40p). Lessers can be crafted into greaters but it's kind of a waste imo. Lenses go on weapons and frames, ideally ones that do a ton of killing. Mine are on Ivara+Redeemer (solo stealth farming) and Excalibur (big mistake that came from listening to some terrible advice, and if I had it to do over I'd put it on something like Equinox or Ash Prime). Have you completed the Second Dream quest? If not, anything else related to focus is a major spoiler. Zenurik and Naramon are the best by far.
Exilus slots are really only needed once you put several forma into a frame, because otherwise you probably are going to fill 74-78 mod capacity with just the normal 8 slots. There's a frame or two that come with a polarity on their exilus though; those I would unlock immediately. Mostly you put Power Drift, Cunning Drift, Rush, or maybe Intruder in that slot. There's not a whole lot else that fits in there that is useful, just bullet jump mods and Enemy Sense for those rare occasions when you can't use your Sentinel/Kavat.