This is a modified copy of my solution for Corporal that reflects the increased effort needed to reach Sergeant rank.
To obtain some of the rewards that you had to purchase in Gears 4 (Banners, Weapons Skins, Character Variants) you need increase your rank. To do this the game has Daily Challenges and Medals. (Click your Left Trigger button while in the menus to see the current challenges)
These are changed daily and include 3 challenges that indicate how many stars you will receive upon completing them. More stars for the more difficult challenges.
Some challenges I have experienced include: Beat a Past Escape map, beat 3 past Escape maps, Complete 12 Eliminations with a certain character (Kait or Swarm Hunter or Swarm Grenadier), Get 16 Eliminations with Assault Rifles, etc....
You can also earn stars by earning medals. These are similar to previous Gears of War games. A number of kills with a certain weapon. Playing so many waves of Horde, a certain number of Escape completions and winning a certain number of matches in a playlist.
Thanks to Futile Youth I now know that you can reroll daily challenges that you have completed, the 1st one is free; then for each after that using your Iron reserves (an in game currency shown at the top right in your Tour page) and obtain new challenges quicker without having to wait the time shown for a new challenge to appear.
The Private Ranks all require 5 stars, the Corporal ranks require 6 stars each; therefore a little dedication and planning with your challenges and medals will help you net Sergeant rank. Just be aware that each Tour of Duty has a reset date, this might not mean much for Sergeant, but you should be aware of the next end date so that you do not work hard to reach a high rank only to have start over a week later.
Good Luck! And have fun!