Fishing is tricky; the goal is to keep the fish in the green section until the bar fills up completely. Some fish tend to hover for a while, others love to jump all over. Best advice I can give is to hold the button rather than tap it repeatedly, and keep the fish near the top rather than the bottom of the green bar; it takes longer to descend than ascend. Also, the green section will bounce off the bottom, so carefully tap to control your speed as you drop.
It's very tricky but once you get the hang of it it'll make sense. As you level your fishing skill up (which increases as long as you catch something, even trash) it will also get easier; the fish move less and your green bar gets bigger. Better rods will help yoo. You'll also unlock Crab Pots at level 3 fishing, which increase your fishing skill as well.
Combat stays pretty simple throughout; there's some items you can get that increase your stats or give you effects like magnetism, illumination, etc., but that's about it. There's different weapon types (sword, dagger, hammer) that have different special attacks (block, rapid strike, and ground pound, respectively). That's about it, it's mostly just to make mining more involved / hazardous than tedious.
Upgrading absolutely helps, but the jump from default to copper doesn't increase power, it only allows you to break the big 2x2 stones. Gold is when you really notice an increase in how many hits it takes to break rocks. Axes are kind of the same way; copper only gives you the ability to break stumps. All the other tool upgrades have very noticeable effects (larger hoe / watering area).