How does one increase one's flag weight if one has gotten over 3,000 points and has earned the privilege to vote? I assume it's bad etiquette to still be flagging instead of using the voting privilege?
Once you reach 3000 reputation, if you attempt to flag a question as off-topic, your flag will get converted into a close vote automatically. A generic "flag for moderator attention" can be used instead, but as you said, that's poor etiquette (and depending on the moderator might result in your flags getting dismissed as invalid, which isn't what you want).
Don't forget, though, that you can also flag answers. If you see an answer that's "not an answer" or is of exceptionally poor quality, you can flag that. There are also the spam and "not welcome here" flags. For that matter, those flags are also still available to you on questions as well.
Once in a while you might see a question that needs to be migrated to a site that doesn't show up in the automatic migration paths. Flagging for a moderator then is completely appropriate.
Last, you can also flag comments by clicking the little flag icon below the comment upvote arrow if you feel the comment is rude, off-topic, too chatty, or obsolete. There's also a general "something's wrong here" flag on comments as well, similar to questions and answers. But comment flags will not affect your flag weight.
With all that said, don't focus on your flag weight and go out of your way to raise it. As you use the site(s), you will come across things that deserve flagging and it will increase naturally.
Increasing flag weight is not a goal in itself. You shouldn't be trying to increase your flag weight, you should be trying to improve the quality of the site. Flag weight is primarily intended at indicating to moderators how helpful you've been to them; it's a very rough measure of how good you've been at performing those cleanup actions that tend to get moderators involved.
Below 10k reputation, the main place for looking for things to clean up is