If you've never encountered a Feebas in a Pokémon game before, that's not surprising. Feebas is not just rare, but it's also incredibly useless as a battler before it's had a chance to evolve. It's unlikely you've run into one on your afternoon stroll through a route, and while you may have battled its evolution, very few if any people are tossing out a Feebas in competitive matches.
However, Feebas evolves into Milotic, which is a beautiful and popular water-type Pokémon that many people use. In most games, Milotic are not findable in the wild, and though they are in Sword and Shield, it's almost certainly going to be easier to capture a Feebas and evolve it instead.
"Easier" being relatively, of course. It's still going to take some time. Here's everything you need to know about getting your hands on one of these elusive fish in Pokémon Sword and Shield.
Where do I capture Feebas?
There is only one place in the game where you can capture a Feebas, and it's going to require some patience (though fortunately not quite as much as it did when the Pokémon first debuted in Ruby and Sapphire). Once you have acquired the ability for your Rotom Bike to travel across water (following the Circhester Gym), return to Route 2 and head to the Professor's house. On your left, there's a big lake. Follow the shore to where you can enter the water on your bike, and get to the island in the middle.
One there, you should see two fishing spots. Feebas will spawn in either of these, and the spots respawn pretty quickly after you fish up something, so you can really just stand in one place. However, Feebas is pretty uncommon. You'll be fishing up a lot of Magikarps and Chewtles while you work on that Feebas, but be patient. Once it shows up, it's very easy to catch with an Ultra Ball or Lure Ball.
How do I evolve Feebas?
You'll need a friend to get Feebas' evolution, Milotic. First, while you're still out on Route 2, where you found Feebas, head around the north side of the lake (still on your water bike) and find the glint in the water indicating an item. There's a Prism Scale waiting for you. Once you have it, hand it to your Feebas to hold.
Now, you'll need to trade Feebas away to get it to evolve. Since you probably want to be the one with the Milotic at the end of this, you'll want to make sure you're trading with a trusted friend who will give you the Pokémon back when you're done.
If you want more Milotics in the future, you can find more Prism Scales on South Lake Miloch occasionally.
Is there any other way to get Milotic?
Milotic can technically be caught in the wild, but it takes very specific conditions. On the Lake of Outrage and South Lake Miloch, Milotic will appear as a strong Pokémon whenever it's foggy out. If you see fog in either of these locations, get on your Rotom Bike, and you should be able to find one. Milotic also can very rarely spawn for Max Raid Battles.
It's best not to bank on either of these though if you have a friend you can trade with, but if you see a Milotic out there, it's good to get it while you can.
Have patience. Feebas and Milotic are elusive, but once you have one, you can breed for more Feebas and make more trades to get more Milotic, especially if you're after the perfect battle Pokémon. Keep hunting, and let us know in the comments if you have any questions about either of these rare water-types.