Brothers are pretty good for beginners… they can come pretty cheap while still being decent machines. Singer is another well-known sewing machine brand, but I have found that recent models are more hassle than they’re worth. My family’s Singer, to put it lightly, was a substandard machine with countless issues.
Sewing is a pretty major hobby for me, so a few years ago my father bought me a Bernina activa 220 and it is absolutely fabulous. It’s far from a beginner’s machine (I just asked my mom, and says it cost somewhere between $700–1000… holy crap I feel spoiled.), but it has lasted me three years with virtually flawless performance. Bernina sewing machines, which are Swiss-made, are bound to be durable and high quality. Their lower priced model, the bernette 56 retails around $199.
Since you’re pretty new to sewing and probably don’t want to invest a lot into a machine at this point, go for a lower priced Brother like the one in your link. Find a model that has as many reviews as you can find – the average of 100 reviews tends to be a much more reliable figure than that of a dozen or fewer reviews.
At this point you don’t need much in the way of fancy stitch settings and such, but make sure it has all the essentials for garment sewing: buttonhole stitch settings, automatic bobbin winding functions, thread cutters, ability to change width and length of stitches and different needle positions. I took a look at the features listed on the Brother LX3125’s page, and it sounds just fine for a beginner.
Have fun sewing!