Like with all Crops, more and more Turnips will slowly spawn on game-worlds where players are active. In the beginning they might be rare, but the longer you play on a game-world, the more Turnips will show up. Staying close to Swamplands for a while might also raise the chances for Turnips to appear there.
The "Pro" world option "more regrowth" should make more Turnips spawn on suitable Biomes on the surface of the game-world even faster.
Turnips can be consumed raw and will then provide 100 additional maximum Health Points for a few minutes, but will not heal player characters. So this is not the best way to put Turnips to use.
Also feeding raw Turnips to Pets is not a very good idea, since nearly no Pet prefers simple Turnips as their favorite Food. And Pets will only provide players with the best pet-harvest when being fed their favorite fodder.
The Cooking Station only provides 4 cooking recipes, but by choosing different ingredients, you can cook up to 36 types of Food with different positive effects (buffs) when consumed (either right click on the Food after putting it into a quickbar, or drag a stack of Food from the inventory over your character's portrait on the right side).
You can grow Crops like Turnips from Turnip Seeds on tilled blocks of Dirt, green Grass and/or Mud. First place such natural blocks over a body of Water or Mineral Water, or alternatively place these liquids directly adjacent to each block of Dirt, Grass or Mud (like directly below for example, or at least corner to corner).
This can even be done underground in complete darkness and/or in a building, but not at too high altitudes, not in too cold nor too hot Biomes/layers, and not too close to Swamplands or pools of Bog Water. In these cases, Seeds will stay "fallow" and won't grow (they can be picked up again though).
Next craft a Plow and use it on the solid blocks to till them; they will turn into "tilled land". Then place Seeds on the tilled land (just like you'd place any block). You can, but you do not need to fertilize Seeds in order to make your Crops grow faster.
Wait a bit until the Crops are ripe to receive the best harvest. Wheat needs approx. 50 minutes (real-life-time) to grow from seeds into ripe plants, even if you're offline. Lettuce needs about 60 minutes, Turnips not much longer (ca. 61 minutes), Horned Melons need ca. 90 minutes until they're ripe.
Crops can be harvested by being "pulled" with the gauntlet and left mouse button (by default) just like any other type of block, item, plant or liquid. If Crops stay "ripe" for 24 RL hours, they will turn into "grown" Crops. The harvest will become a little less then, but Crops won't wither.