Togepi is one of the cutest Pokémon in the series, and you can find one Pokémon Sword and Shield. While it can stay cute for as long as you have, Togepi does evolve. It has two evolutions, Togetic and Togekiss.
Before you can think of climbing Togepi's evolution tree, you first need to find one. You can only find them at the Bridge Field location in the wilderness during normal, cloudy and heavy fog weather. Again, the chances of locating one are slim, so you may find yourself wandering around this area for a long time.
Unfortunately, Togepi doesn't show up in the overworlds and is instead one of the exclamation points that force you to run towards them in the grass.
Fortunately, a quick way to get a Togepi is to talk to a woman west of Hammerlocke. It will be in the left corner, where Rhydon and Gastrodon are fighting. She talks to her and is willing to trade a Togepi with a Toxel. If you manage to acquire a Toxel, you can receive Togepi much faster than just research.
When you get your hands on Togepi, you need to build your friendship with the creature. You have several ways to do this, with the simplest having it at your main party while traveling the Galar region. You can also feed him curry in your camp, provide him with berries that increase friendship, or have him take part in Pokémon battles, without him passing out. If he passes out, you lose some friendship.
You can check your progress with Hammerlocke's friendship check. When you have a high enough friendship, you need to level it once again for your Togepi to become Togetic. Togepi starts out as a Fairy type, but changes to Fairy and Flying type after he becomes Togetic.
Now, to acquire the final evolution, you need to use a shiny stone on your Togetic. You can find the polished stone in only a handful of places in the Galar region. There is one behind the large stone circle on Lake Outrage, or you can try your luck with the Digging Duo to have them dig one. You can find them at Bridge Field, near the nursery. It costs 500 Watts for their services.
If you have trouble acquiring a polished stone, there is a chance that Clefairy, Clefable and Lunatone will hold one when you capture them. You can cultivate the respective positions to prove your chance of getting a pebble this way.
When you find the stone, give it to your Togetic and it should evolve into a Togekiss shortly after.