You said it, it's a sandbox game. Do what you want, there are a lot of different ways to survive 100 days.
Since you specifically asked for the first 20 days (and winter hits on day 21, so that's convenient), here's a small checklist for your pleasure. Feel free to edit in stuff you think is missing:
Days 1-5: gathering
Explore as much as you can, gather food to stay alive, and some basic resources:
- Twigs, Grass
- Flint, Axe, Logs, Pick, Stone, Gold
- Science Machine, backpack, spear
Find a balance between picking up just what you need & picking up everything you can. You'll want to cover some ground.
Days 6-10: settling
Building a small camp to ensure food production. You'll be needing manure, so close to beefalos (or a pig camp) is advised.
- Science machine, fire pit, crock pot, drying rack, farms
- Lightning rod, shovel, berry bushes, saplings, grass tufts
- Chests, log suit, football helmet (if you can).
Days 10-15: improving the camp
Explore the lands around you, bring the good stuff to your camp:
- Bee boxes, flowers, fireflies.
- Survive your first hound attack.
- Dig up graves for the pig king. Find gears and create an ice box.
- Grab some pig skins and build a few pig houses nearby.
Days 15-20: prepare for winter
Winter is coming. It will be cold, you'll find it harder to find food & sanity.
- Shave beefalo, kill spiders, collect rabbits. Make winter clothing.
- Cut rocks, make a heat stone.
- Gather honey, taffy, jerky. Craft a top hat. You'll need the sanity.
- Sleeping roll or tent to convert excess food into easy sanity (and skip boring nights)
- Drying rack, in order to make meat take much longer to spoil and give sanity