First, in most cases resigning so shortly after starting is unlikely to go over well, unless they're thinking of firing you anyway but that doesn't seem to be the case here. Additionally, it is much, much easier to find a new job when you're currently employed. It signals that you can work in an office environment, you're building experience and you're usually less stressed during an interview and come across as les desperate.
The saying you mention corresponds to the idea of a "dream job" but unless you are financially independent you are not going to find that. Every job will have downsides or require you to spend time doing stuff you'd rather not do. That is not to say that you can't like your job and you want to find one where the good (vastly) outweighs the bad, but put the idea of a dream job out of your mind right now: you'll be a lot happier.
Given that, my advice is to just tough it out for now. Frankly speaking, you give the impression that you simply need to adjust to working life. Unfortunately your halcyon days of college are over and the sooner you accept that the better. You're entry-level so depending on your field you're very likely to be stuck with low-level work until you build experience and prove to management (or potential new employers) that you can perform well at more demanding and interesting tasks.
In closing, your title question is misleading: in this job climate and given your statement that there are no entry-level jobs in your field you're very unlikely to find a new position within several months that you'd actually enjoy. I don't mean to discourage you but you need to be realistic about what jobs are available to you at your current level of experience.
To tackle your add-on questions:
How do I ethically resign after working less than a month?
There's no ethical problem here, you're employed at-will. Like I said, you risk coming across as dishonest and burning a bridge but this kind of blip is generally short enough to leave off your resume without much impact on your hiring chances. If word got out you risk being marked as a job hopper if you turn it into a pattern (multiple jobs of less than a year in length within the span of 2-5 years).
Is it ethical to apply for other jobs while currently employed?
That's what everyone does. In the US people usually don't even announce that they're looking for fear of reprisal. Employees usually announce that they're leaving two weeks in advance which is called their notice period. If you have a contract (most US employees don't) it might state a different notice period so keep that in mind.
Do they hire people who are already employed?
As I mentioned above it's much easier to get hired if you are currently employed.