Once you have a decent line of nets, it sort of calms down. One tip I’ll offer for food involves something you can build before doing any of the story islands, but I’ll spoiler it in case you want to keep totally blind:
Craft a cooking pot (table) and then start making vegetable stew, which only requires any combination of four potatoes or beets. 2/2, 3/1, 4/0, 0/4, whatever. Once you’re hungry enough, eat two in a row and you’ll be full for quite a long time.
Oh, that reminds me, this I won’t spoiler because it’s not really a mechanic that is ever explained: The hungrier/thirstier you are, the more effect you get from food and drink. That means you shouldn’t top off, but rather wait till you get woozy (or almost woozy) and then fill that bar up near full in one go. There’s also a “bonus” mechanic later on, which you can check out if you follow my spoilered tip above. It’s more relevant for food than drink.
As for your general question, you can make it relaxing, yes. You’ll need some stuff from the story before you can carve out significant downtime, but it does get there. It’s funny, though, ever since posting the link to my full game above I started rewatching it from the beginning and got totally sucked in. I’m a little over halfway through now, and in these videos I never slow down for a second. It’s like 17 hours of non-stop manic action, sometimes doing two things at once. (While fishing, you can still do some things like empty the nets.)
Tips for Bruce the shark: If you destroy the thing he’s biting with your axe, you get some of the materials back, making it cheaper to rebuild. Costs wear and tear on the axe, of course. You can also remove a net he’s biting and just put it right back. They don’t take damage. Killing Bruce is also a decent source of food.
EDIT: Here’s some intro tips for drink, again available before you even start the story: Race to the advanced water purifier and craft yourself a bottle, not the cup. Both the advanced purifier and the bottle hold five cups of water, and the purifier works for free since it’s powered by the sun instead of a wood fire. (It still works if you cover it.) Having five drinks in a container is a huge improvement over the initial purifier and cup. Combine this with the food tip above to really take the edge off survival maintenance. One of those items requires vine goo, which you get by smelting seaweed.