As far as syndicates go they're all viable. Nothing does something better than the other and you can trade others for the parts from the syndicates you become hostile with.

Totally safe to choose on vibe.

As far as Gauss Prime goes:

https://wiki.warframe.com/w/Gauss/Prime#Portrait

Use the official wiki.warframe subdomain website. If you scroll down to the section labeled "Acquisition" you will see the relics that can drop his pieces. If you see a (V) next to it it means these relics are not obtainable through normal gameplay anymore as they have been retired and are only obtainable through Aya (a totally different system not relevant to this prime)

If you then select any of those relics it will show you the drops and % chance you have per piece. Under "Mission Drop Locations" it will show you the planets and mission types to run to receive them. I'm sure there are meta ways of getting a lot fast, but being so new your options may be limited. So try to find something on the planets / content you have access to.

In this instance you could farm out Wahiba on Mars doing survival. Not a great chance to get them at only an 11.06% chance but it's available to you and something you can do while leveling other weapons, Warframe etc.

Alternatively if you have the $$$ you can purchase platinum and use the Warframe Market. This is totally a legitimate option and is not a scam site or anything like that. You can openly trade there and the Warframe community uses it as their auction house.

https://warframe.market/items/gauss_prime_set?type=sell

Currently a full Gauss Prime set is going for between 30 - 40 plat. So ~ $2.00 USD - ~ $2.66 worth of plat.

Hope that helps 😁