What tasks should I try to accomplish in the first 20 days?

Clearly Don't Starve is a sandbox game, and you have quite a bit of freedom to play the game as you please. (Quite frankly, I've spent a lot of the time wandering around the game fairly aimlessly.)

But, I'm wondering if there are certain tasks that should take a higher priority within the first 20 days.

Clearly, building a Science Machine and/or an Alchemy Machine are very high priorities. And, next I typically have been trying to acquire some warmer clothing.

But, where should I be focusing my attention from here?

Should I be building structures? Farms? Exploring the landscape?

Basically, I'm looking for a Quick Start guide to having a successful long game.

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.
• Boomerang.
• 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