I already have both the science and alchemy stations, and a stone fire pit. What else should I add or research to make a stable base?
Well, most of the resources that are renewable and portable you will want nearby. That means using your shovel to dig up bushes, saplings, and tufts of grass and planting them all nearby. Plant trees nearby as well. Ideally these things will be close enough you can get to them at night to have something to do before the sun comes back up, but you don't want to catch them on fire.
You can add farm plots nearby. With farm plots as well as berry bushes, you should have a pretty stable food source even without having to leave your base, or at least without having to leave very often.
You'll want to be able to get manure -- this means either building your base near a beefalo herd (good because they don't ever go crazy and try to kill you, but they wander) or a pigman village (good because you'll always know where to find them, bad because sometimes they go crazy and try and kill you, plus you have to give them stuff in return for manure).
You may want to be within range of a few other resources. Pigman villages you can construct, spider eggs you can move. Either of these, if you want them, should be placed a few screens away so you don't have to fight through the night. Placing your base within a comfortable walk of some tallbird eggs might be a good idea.
I placed my base near a beefalo herd (which is now spreading out pretty well, so this is becoming less effective) as well as a bunch of rabbit holes. I don't have easy access to tallbird eggs, but between farms, bushes, rabbits, and enough manure to keep my farms and bushes going, I pretty much never have to worry about food or basic resources (sticks, wood, grass).
Other than that, just keep things near your base once you can research them -- crock pot, chests, etc. The things you keep closest to your fire should be the things you want to mess with overnight. Hostile (or potentially hostile) things should be kept a few screens away. Everything else can go somewhere in between.
I find it easiest when you make lots of stone walls and surround your base in a grid formation, screen by screen. Try to make several entrances, and put as many Tooth Traps in the hallway as you can. Make the hallway at least fifteen stone walls. You’ll need the following things if you want a “Never-leave-your-base-ever” lifestyle:
Pigs for help and poop. (Help is only needed if you want to leave your base for a questing adventure, like a cave. Make sure to bring lots of meat or jerky.)
Traps for rabbits and such (next to rabbit holes).
Beefalos for fur and horns (not necessary, but nice).
Put Lightning rods, crock pots, chests, drying racks, a birdcage, and stuff you want to tinker with near your fire.
Fire pit. Nuff said.
If possible, get the “King of the Pigs” in one of your “Screen Squares”.
Berry Bushes, Grass Tufts, Saplings, Trees, and Farms in a screen square.
Tent(s) (again, not needed, but nice)
Lots of sanity reducing and refilling items, such as assorted mushrooms (to get Beard-lings and such).
-Get ready for winter, even when one has just ended. Always stockpile food, eat as little as necessary, and when your starving, make some friggin Meatballs.
-If you are starving, eat a cooked meal, not the raw materials. Remember: Would you rather eat a Small Jerky, or some Meatballs?
-Some of the best cooked meals are (In my opinion) Fistful of Jam(Put Berries in all the slots.) Meatballs(One meat at the top, and three vegetables under it.) and a Bacon and Eggs, witch is made from one egg at the top, two meat below it, and a stick at the bottom. Keep these foods and others in your base to ward off starvation.
Get two Heat Stones (one to leave at the fire, one in your pocket, and you can switch out when necessary).
Make sure you’re never at the pig base at night. Don’t risk it!
And remember, your squares don’t have to be a screen size, or even squares at all.
The most important rule: Don’t ever starve!