Minwax water-based wood stain is a good product that should provide you with a nice finish. Staining is a bit of a tricky process, and you might have to live with a few imperfections, but it's very rewarding to completely transform a piece.
We got a pint of this Minwax stain on an unbelievable sale at Menards. It was on the shelf with other paints and stains that had been made up with custom colors that the patrons then apparently didn't want. It was the perfect blue shade I'd wanted for our guest bath, and we got the pint for $2.
It's a good idea to buy the base coat for this product, which evens out any imperfections in the wood. Since this was for a bathroom, we also bought a polyurethane top coat to protect it from water damage. It definitely is a lot of work, and takes a few hours for each coat to dry. With the base, stain, and top coat you'll need at least four coats total. Definitely work with plenty of ventilation and a fan nearby to help circulate the air. (Make sure the fan is clean so you're not blowing a bunch of dust into the air!)
I applied the stain and then rubbed it off again with a cloth to give the wood an antiqued look. I worked on the back of the doors first to practice. When doing cabinet doors, be aware of drips along the edges that can unevenly stain the opposite side.
The Minwax stain goes on very easily with a foam brush, but you can also use a synthetic bristle brush or cloth. It all depends on what look you're going for. Additional coats make the stain darker and add richer color. There will be some variations because of the natural wood grain, but that's part of the beauty of the final product.
I was pleased with the final result, and loved that the cabinet could have such a nice color to it. We've had the cabinet stained now for three years and it looks just as good as it did when first finished.