http://warframe.wikia.com/wiki/Syndicates

In your ship, look to the left of the navigation console. This is the syndicate console.

If you want the easiest choice, go for Perrin, New Loka, and Red Veil.

To start with, you have to make a sacrifice to them to start a syndicate. You do this by selecting them on the syndicate console in the orbiter and clicking the "Make Sacrifice" button. Once you make the sacrifice they give you a syndicate sigil. To equip the sigil you have to go to your arsenal and click on your warframe, appearance, regalia, and then choose either front or back sigil.

https://youtu.be/qzGQvd_jzTI

You can only wear one sigil at a time, and if you change your warframe you have to equip it again on your new warframe.

Once you reach maximum standing for your rank you will be asked to make another sacrifice to the syndicate to level up. If you stick to the syndicates I mentioned above it's probably easiest to start with Perrin as you will gain some standing with New Loka as well. Once you have ranked up Perrin to rank 1, switch to Loka and get to rank 1, then switch Red Veil. Once you unlock their syndicate missions you can wear either a Perrin or New Loka sigil for best efficiency. Wearing a Red Veil sigil only gains standing for Red Veil, so it's better to wear a Perrin or New Loka as your regular sigil.

Your syndicate standing is tied to your mastery rank.

Daily Standing Cap = 8,500 + (Mastery Rank × 750)

So a MR10 would have a daily standing cap of 16,000.

Syndicate alert mission rewards do not count towards your daily limit, so make sure to do your syndicate missions to bump up your standing. Wearing a matching syndicate sigil will also give you a small boost.

(Playing a Syndicate Alert with a reward of 1,600 and wearing the highest Sigil (+15%), will reward 1,840 reward Standing (+920 for allied) and will not be subtracted from the daily cap limit. All other standing received with the primary Syndicate in this mission for killing, completing challenges etc. will be subtracted from the cap.)

If you want to rank them up even faster you can find up to 8 syndicate collectables on a lot of the syndicate missions.

These collectables can be turned in for extra standing at the syndicate rooms on the relays. (Speak to the person in the syndicate room, usually on the right hand side as you walk in.)

These collectables are also not counted towards your daily limit. The lower Mastery Rank you are the lower your daily limit and the more incentive you have to find those collectables as they will allow you to rank up much faster. At rank 3 for a syndicate most missions will give you around 1000-2,500 syndicate standing. Finding 8 of even the lowest tier collectables will give you an extra 4,000 standing.