Buy a push or pull cart. I carried for a few years before I switched to a hand cart and I wish I made the change earlier. Carrying a bag (vs a hand cart) is more tiring, damages your clothes, and puts additional strain on your back.
As for tips on the actual walking aspect, learn the optimal places to put your bag when you're around the greens. You can't cover as much distance on foot as you do in a cart, so placing your bag along an efficient route between holes is beneficial.
A lot of people will leave their bags far from the tee if the tee box is out of the way of the walking route. If you do that, make sure you bring enough clubs for whatever shot you may need. Carry an extra ball and tee in your pocket in case you hit an errant shot so you don't have to walk some 30 yards back to your bag for a provisional ball. I actually hate doing this, so I'll drag my cart with me all the way down to a tee box, even if it is out of the way.
Be ready for your shot faster. This one isn't really much of an issue because you're not sharing a cart with anybody, so while others have to drive around to different places, you can just walk directly to your ball and wait for your turn. While you wait, you can size up your lie, get your yardages, and prepare for your shot. Everything that isn't your preshot routine can be done ahead of time.
Pay attention to your ball. You don't have the luxury of zooming along your flight line and driving in circles around the general vicinity of where you think your shot landed to look for it. Try to follow it as much as you can and narrow it down to specific landmarks. Don't just see it go towards the woods and say "it went by the woods." Narrow it down as much as you can, "it came down at the third tall tree from the left." This will speed up the process when you have to look for a ball.
Bring plenty of fluids, and snacks if you have to. If you are just starting to make the switch, you're probably going to underestimate how much energy it takes to walk and carry 18 holes. An extra towel is good too, if the weather is hot and you sweat.
If I can think of anymore, I'll add them.