Apologies for the crummy illustration, I'm a graphics doofus :), and for not providing full instructions for the basic joints I'm mentioning (Google!)
I would start by making the "A" at each end with a half-lap joint (at the parallelogram where the legs cross - Google "half-lap joint".) You can screw them together with four screws inside the corners of the p'gram, or clamp and glue together. Then cut a through mortise (another thing you can Google) through the middle of the lap joint you made in the first step (shown as the little rectangle in the parallelogram.) You can do this with, at minimum, your drill and a good sharp chisel. Once you've cut the mortise, fashion a tenon (Google it!) at the end of the cross bar piece that fits snugly into it. Use your saw for this, cut close to the line, then clean up with your (new) chisel. Test fit, add glue, and assemble. I think this would give a nice solid joint.
Agree with @Matt, go with larger wood, like 2x3. I would like to note since you're a beginner that glue is stronger and more permanent than screws, and probably safer here because the screws could loosen over time, especially in something with movement like a swing, and pose something of a hazard (swing fall apart/down.)