Mathematical journal for an undergraduate

I think the three best possibilities here are the American Mathematical Monthly, Mathematics Magazine, or the College Mathematics Journal.

A table of contents and some info on AMM can be found here.

Similarly for MM here. And the CMJ here.

Of the three, I'd estimate the CMJ to be the most approachable. But a large idea of it is aimed at educators of upper high school or beginning to mid undergraduate math. AMM and MM are both approachable, but will also have some things which likely aren't familiar to an undergrad. But every article is written in an incredibly approachable way.

All three have problem and answer sections as well, with the general difficulty being AMM > MM > CMJ (least difficult). Nonetheless, it would be rare for an undergrad to be able to do all the problems every month. Let me emphasize - really rare.

Perhaps the most important aspect is that the math articles and papers, which I consider to be very real, are not terse. The articles are written to be approachable and dense, often with large doses of humor. I used to read the AMM and the MM all the time, and I enjoyed that part of my undergraduate days. I still write up a solution to a problem every now and then, too.

Have you considered the Mathematical Intelligencer? The Mathematical Intelligencer publishes articles about mathematics, about mathematicians, and about the history and culture of mathematics. It informs and entertains a broad audience of mathematicians and the wider intellectual community. The Mathematical Intelligencer welcomes expository articles on all kinds of mathematics and interdisciplinary trends, and articles that portray the diversity of mathematical communities and mathematical thought. Humor is welcome, as are puzzles, poetry, fiction, and of course art. Forthcoming issues will feature emergent mathematical communities around the world, new interdisciplinary trends, and relations between mathematics and other areas of culture. Makes a great reading!!

The MAA journal Math Horizons, which comes out four times a year, is explicitly geared toward undergraduates.