I use smartystreets goconvey and assertions packages regularly. While the stdlib offers most everything you need, one thing I think it lacks is "quality of life" features.
The goconvey package will provide you with a nice web browser interface with hot reload. Everytime you update, it re-runs. You can also selectively choose what runs. So I typically open another console with it running which pops open a browser tab, as I code. It also makes looking at code coverage easy and convenient.
Organizing and nesting your test cases is also made easy and then renders nicely in your web browser.
Then of course the assertions package just provides you with some shortcuts.
What I've found is that this saves a lot of boilerplate and time. I'd be writing all the same stuff for each project otherwise.
It also makes your tests more consistent. If you have a library of assertions and tools, you will reach for one there first before building a custom pattern that you then deviate from in another test. It's not hard to end up with messy, hard to read tests.
There's other packages that help with testing APIs and HTTP requests as well. So like others said, you can do it from scratch, or you can use a package that takes care of the boilerplate. It could leave your tests shorter and cleaner.
So "what kind of tests?" All of them pretty much in my experience. I've always benefitted from other packages. Though the important thing to keep in mind is that all this is still built on top of the already awesome testing available in Golang. One reason why I like Golang over other languages is because it has most everything baked in already. I'm not bolting on something that's missing, I'm simply enhancing it for my work flow and preferences. Then the next person who comes along to run tests for my package is actually free to use just the CLI of they want. No need to run goconvey in their browser. So the preference parts stay preferential. This is not quite so common elsewhere.