How you dress for your fishing trip depends largely on where you are going and what the climate is like there. If you are just going to mosey on down to the local pier and drop in a line, all you need is your hat and sunglasses. Oh, and some sunscreen if it's warm enough to wear short sleeves. The Aussies have a saying to help combat skin cancer; "slip, slop, slap." That means slip on a shirt, slop on some sunscreen and slap on a hat. That's a good way to remember skin safety and it's not just for sunny weather; you can still get sun burnt on a cloudy day.
Sunglasses will cut down the glare to your eyes and help you to see through the water down to where the fish are. A hat or a cap with a good brim will help to shade your eyes as well as your face. If you are fishing from the beach, the heat reflected off the sand can still burn your face even though you have a hat on, hence the need for sunscreen.
Of course it may be freezing cold where you intend to fish. In that case you'll certainly need warm clothing to protect you. No use catching that big one if you die of hypothermia before you get home with it. A light rainproof jacket would also be a good thing to take with you, especially if you intend to fish from a boat or from rocks. Sea spray can be mighty cold on a cool day and you never know when a sudden rain shower might hit. If you are out in a dinghy, there will be no shelter to run to.
Fishing from a boat also comes with more risk due to sudden rough conditions that may well spring up before the fishing trip is over. If you are adequately clothed, you won't have to cut your trip short due to weather conditions, unless of course, it becomes dangerous. If you are caught out in rough conditions and the boat overturns, a lifejacket could save your life. In fact in some areas lifejackets may be compulsory, so make sure there is one for each person in the boat.
Of course, ice fishing will require the warmest clothes you can get, and you may need to buy them to make sure of getting the right ones. Boots are particularly important to prevent frostbite. If in a boat, you'll need footwear with good traction on the soles to prevent slipping, while waterproof boots are necessary if you intend to wade into shallow water.