1. Learn how far a normal jump will take you. There's a surprising amount of acceleration mid-air. Practice jumping in prior zones to get a feel for how far a single jump can take you.
2. Learn to pull back when you want to stop. Because Sonic carries a lot of momentum, you'll need to actually press the opposite direction to slow down quickly enough when platforming sometimes. It's not enough to just let go of forward... you need to actually press backwards sometimes.
3. It might even help to turn OFF auto-brake while you practice in safer zones like Greenflower and Techno Hill, since it will force you to pull back more often, thus training your muscle memory for situations where auto-brake wouldn't be enough.
4. Remember to hold down the jump button for maximum height. Some jumps can only just barely be cleared by Sonic.
5. Try a different route. Castle Eggman has several different pathways (some of which are a bit hidden), so if one route is too difficult, try finding another.
6. Learn how the chain swings work. When Sonic is at the bottom of the swing, he'll fly straight forward. 90 more degrees, and he'll fly straight upwards. The current direction Sonic is moving at the moment you jump off the swing determines the direction he'll go. Picture it like this: the moment you press jump: the swing disappears, and Sonic will keep traveling in the direction he was moving at that exact moment.
7. Remember that you can use the thok to boost and/or redirect your horizontal movement during a poorly-aimed jump.
8. In sections with the Robo-Hoods (the green guys that shoot the arrows), don't stand still for too long. Always be moving.
9. The downward plank with the swinging maces can be cheesed by just jumping and thokking over the whole thing.
10. Before entering a section with multiple moving platforms/hazards, take a moment to observe the pattern so you can get a feel for how you will need to time your movements.
11. When using a diagonal spring, don't press any direction on the movement keys/stick, unless you really know what you're doing. The vanilla campaign is designed in such a way that diagonal springs generally don't require additional input, unless perhaps you're trying to use a spring to reach somewhere other than the standard landing spot.
12. In contrast, vertical springs pointing straight up require you to run into them in the direction you want to go. There's one notable case of this in CEZ1. Remember that the speed with which you enter the spring determines how far you will go horizontally, and remember what I said earlier about pulling back to slow down.
13. SRB2 has tons of options to adjust how the controls work and how the camera behaves and where it is positioned. I can't recall what SRB2 uses by default, but if you're using a gamepad: I recommend picking something that puts spin and jump on the bumpers, thus freeing your right thumb to always be on the analog stick for the camera.
14. Zooming out the camera might help with spatial awareness, at the cost of depth perception.
15. Remember that Sonic has a shadow which always points straight down; this can be used to help determine his horizontal position.
16. The Software renderer currently looks a lot nicer (in my opinion), but the OpenGL renderer works better with things like orbital camera and more-vertical camera positions.
17. Using 3D models for the player character may also help with depth perception, and this will once again require using the OpenGL renderer.
18. A lot of the fun in a game is finding all the secrets by yourself, but if, after applying all of the above techniques, you're still struggling, consider watching a video or two of someone else playing the zone... preferably *not* a speedrun, since those tend to employ techniques far beyond the capabilities of novice players.