While swimsuits are normally sewn with a serger, I've sewn bathing suits for myself before on a straight stitch machine. If you only have a straight stitch, long-length, narrow-width zig zag stitches are your friend, as is gently pulling your fabric as you sew, so that your seams will stretch. Most machines, if not all, can accommodate twin needles. This is also an excellent way to achieve a stretchy seam without a serger. I am a firm believer in making do with what you've got at hand!
You are going to start by layering your lining and self pieces, wrong sides together. You will treat them as one piece. Line up your side back and centre back piece, and stitch together, 1/2" away from the raw edge.
You will assemble the front in the same manner. Layering your self and lining, treating as one, and sewing side front to centre front, ending up with this:
Looks nice and neat on the inside, huh?
Flat lining is the easiest way to line a garment, since you just treat your lining and self as one piece. I like this method for lingerie and swimsuits because it prevents any shifting between the lining and outer layer, which is important on garments that are close fitting. A shifted lining can cause lumps and bumps that show through.
Once you have completed sewing the side fronts to the centre, you can stitch your bust cups on. I start my stitching line from the centre. For me this is the best way to keep that nice sharp point where the cups meet in the centre!