Surfing works a bit differently in Pokémon Sword and Shield than it has in previous games, but as ever it's still just as important. Surfing allows you to cross water and catch Pokémon you are unable to run into on land. Because this progression is story-related, you won't miss out on the ability to cross water.
How to surf in Pokémon Sword and Shield
Source: iMore
Instead of teaching a Pokémon the move Surf, you'll now be able to cross water once you obtain the appropriate bike upgrade. This occurs after defeating the 6th gym leader in Circhester and making your way down to Circhester Bay on Route 9. You'll notice two Team Yell members hassling another NPC. After defeating them in battle, the NPC will reward you with a bike upgrade that allows you to travel across water.
All you need to do to use this upgrade is simply be riding your bike. It will automatically switch to the floating bike as you transition from land to water. You can't fish while on your bike, however, so you'll need to go back on land and stand by the edge to use your fishing rod.
Can I still find the HM Surf?
Yes, but it's not called an HM anymore. In Pokémon Sword and Shield, Surf is a TR, meaning you can teach it to a Pokémon but it's only a one-time use unless you buy more of them from a trader in the Wild Area. Only TMs in Pokémon Sword and Shield can be used a limitless amount of times. You can check your bag to see which moves are TRs or TMs, and how many TRs of a certain move you have.
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