How to maintain a straight cutting angle with a tenon saw?

Now I want to make fine cuts so that the legs fit into the apron, how can I ensure my cuts are straight?

Using a hand saw for fine work is a skill that takes some time to develop. But furniture makers did all kinds of crazy stuff by hand before power tools (or even power!) were available, so it can be done. I can't claim to be that good myself, but here's what I'd do if I were going that route:

0. Use the right saw. Tenon saws can be set up for ripping, i.e. cutting along the grain (which is what you do when you're cutting the sides of a tenon), or for crosscutting, cutting across the grain. If you're cutting a workpiece to length, you probably want a crosscut saw.

1. Mark your cut lines. Use a marking gauge, preferably one with a cutting wheel instead of a point, to mark a fine line where you want to make the cut. Figure out which side of the line is the "waste" side and draw an X there. Your goal is to make a cut that lands exactly on that line, but the saw kerf has some width, so you really want the side of the kerf to be on that line. The marks, by the way, should be on the top and front and back sides of your work, so that you can easily track both the side-to-side ("yaw") and vertical ("roll") angles of your cut.

2. Use a guide. Clamp something with a flat, vertical face to your workpiece so that the face lines up with your line. This gives you a handy reference surface that you can run the saw against. It'll help you stay on your line and train you to hold the saw vertically at all times.

3. Cut slowly. It's easier to nip a problem in the bud, so go slowly and keep a close eye on your line. Stop every few strokes when you're starting out and look at how you're doing: Are you on the line? Are you keeping the saw in line with your guide?

4. Pay attention to your form. I'm no expert, so I won't try to give you too much advice here, but it's safe to say that you want the arm holding the saw to be moving in line with the cut, so your shoulder and elbow should both be lined up on the cut line. Even more important than that, though, is to be aware of what you're doing and how that's affecting the cut. If you find that you get better results some other way, do what works.

5. Use a shooting board. For really clean results, consider cutting just to the side of your line and then cleaning up your cut with a plane and shooting board, or (if cutting tenons) a shoulder plane. If you haven't seen one, a shooting board is just a panel that has a fence to support your workpiece, and a surface that your hand plane rides on on to cut at a fixed angle to your workpiece. It's a basic piece of kit for anyone using hand tools. Check YouTube for videos about how to make and use one; it's not complicated.

6. Practice, practice, practice. Those folks who can make a clean cut on a line without a guide and without stopping didn't start out any better than you, they've just spent a lot more time working at it. The more you do it, the better you'll get.