Read design blogs and publications, visit art / design galleries, and go to museums. A person’s design is influenced by the designs they surround themselves with, and if you’re subconsciously channeling / filtering / replicating aspects of award winning designs, yours will become better.
Bear in mind that the web is just another medium for communicating visually. In many ways it borrows from the paradigms of print design, which borrows from the paradigms of fine art, and so forth. So in other words, don’t limit the things you influence yourself with to only web design.
Along those lines, drawing (on physical paper) will equip you to tackle visual communication problems more creatively and effectively because it’s more limiting than working on a computer. It’s like having a single stick and a single drum to make a rhythm versus having the world’s most elaborate drum set in front of you and twenty arms.
Find the most “frictionless” tools for you. I added this in because you mentioned designing in the browser which can be effective and in some cases save time, but you’re at the mercy of what your coding abilities allow you to do. However the same can be said for tools like Photoshop or Illustrator: if you don’t know how to operate smoothly in them, you may find yourself spending more time reading the manual and following tutorials than actually designing. Ultimately you want to be able to change things quickly in order to quickly experiment and see what they look like rearranged, resized, etc.
Good design is about removing the unnecessary and adding just enough to get the points across.
Remember that there are always multiple designs that would work, and seek them all out. Don’t get married to a single idea, especially early on.
Give it time. Design something, then walk away for an extended amount of time (a few hours at least). Come back with a clear head. You’ll be able to see it more objectively, and will notice things about it you overlooked.
Find what inspires you outside of visual communication and use it to recharge while you aren’t designing. I love films, video games, and music, so when I’m not working, I try to engage myself in those things. I find they inform my designs in subtle ways, and since they’re creative mediums as well, there can be interesting crossovers.
Run your work by people who you feel are competent designers that would give constructive feedback. It doesn’t mean that you have to take their advice, but having it in mind is helpful and they will notice things that you may not have. Also learn to take criticism gracefully, and find your own balance between defending your design and giving in. You don’t want to be a pushover, but you don’t want to be an asshole.
Find a mentor. Ideally this would be someone with whom you work that you work with on a daily basis. I find these relationships just naturally happen, but it goes back to number nine above where you may need to seek them out more than they seek to give you unsolicited feedback.