To enter a Season, become a Specialist by reaching Tier 4. Qualify for the Season reward by winning 3 battles, and you'll be able to collect your reward at the end of the Season.
Keep track of your Seasons progress at the top of the Tiers menu. Keep winning those battles to get a better reward!
There are three additional Prestige Tiers starting at Master, Tier 8. At the end of the Season any additional Trophies earned in these Tiers are converted into Prestige Points, a permanent score for your profile. Your Trophy count is then reset back to 4000 when the next Season begins.