Despite some content being available with the Vanilla game and no DLC's, I recommend getting Snowfall before using any of these.

I created a 13-month calendar where March 21st is the New Year and every month is exactly 4 weeks or 8 days long, with the exception of the first month every year with 29 days and the last month every leap year also with 29 days.

Ephraim: March 21st-April 18th.
Reuben: April 19th-May 16th
Simeon: May 17th-June 13th
Levi: June 14th-July 11th
Judah: July 12th-August 8th
Dan: August 9th-September 5th
Naphtali: September 6th-October 3rd
Gad: October 4th-31st
Asher: November 1st-28th
Issachar: November 29th-December 26th
Zebulun: December 27th-January 23rd
Benjamin: January 24th-February 20th
Manasseh: February 21st-March 20th

OR...you can stick with the Gregorian Calendar and say March-May is Spring, June-August is Summer, September-November is Autumn, and December-February is Winter. But to use the above calendar, here are the instructions:

1. When starting a new city, make sure you know exactly if the Groundhog saw a shadow or not. If the Groundhog did see a shadow, the three seasons for temperate zones are as follows:

Winter: Manasseh, February 21st-March 20th
Spring: Ephraim-Simeon, March 21st-June 13th
Summer: Levi-Dan, June 14th-September 5th
Autumn: Naphtali-Asher, September 6th-November 28th
Winter: Issachar-Benjamin, November 29th-February 20th

If the Groundhog did not see a shadow, your three seasons come a month early as follows:

Spring: Manasseh-Reuben, February 21st-May 16th
Summer: Simeon-Judah, May 17th-August 8th
Autumn: Dan-Gad, August 9th-October 31st
Winter: Asher-Benjamin, November 1st-February 20th

If you're starting a tropical city, then note the following:

Pre-Monsoon: Ephraim-Simeon, March 21st-June 13th
Monsoon: Levi-Naphtali, June 14th-October 3rd
Post-Monsoon: Gad-Issachar, October 4th-December 26th
Dry: Issachar-Benjamin December 27th-March 20th

2. Keep in mind the very date you're starting your city IRL. That is precisely the same day it will be in-game once you start your city. If the Groundhog did see a shadow, use the first chart above to determine which season your city will be in at the start of the game. For example: if today's date is August 25th AD 2019, and the Groundhog did not see a shadow on February 2, AD 2019, then Manasseh was a spring month and not a winter month, which means that Dan, being mostly August except for 5 days of September, is actually an autumn month. Therefore, I would use Autumn theme for my city as it starts, and that Autumn would last until All Hallows Eve, or Halloween, or October 31st, or Gad 28th. Once that date hits, I would pause the game, save it, and reload the game using Environment Changer mod in Winter. That winter would therefore last until either the end of Benjamin, or February 20th, or the end of Manasseh, or March 20th, depending on my dice challenge result when the in-game date says February 2nd.

3. Using Climate Control, switch to Simulation Time and watch the in-game date on the bottom left corner of the screen. Simulation Time will cause that simulation date to determine the temperature, climate, and weather of the city. When you come to the last day of the season, pause the game and save it so you can load into the next season.

4. So, I mentioned Groundhog Day many times before. As a Dungeons and Dragons nerd and a math enthusiast, I declare Mini Miny Mo completely pointless! Flip a coin! Roll any die for Odd or Even! Play a quick game of Craps, or Rock Paper Scissors with a friend, for cruising out loud! Miny Mo is too predictable! This will determine if the Groundhog saw a shadow.

Believe it or not, for a simple math analysis of having 13 equal-length months in which the first day of the year is March 21st, I find that the results made a lot of coincidental sense, so I became a big fan of it.