Destroy the Machines
This is truly the name of the game in Sonic CD. It's not required necessarily to go back in time, but it's recommended that you do. As we talked about, the game allows you to explore the past and future of each level, essentially giving you three different designs of each level (including the present).
It's a good idea to travel to the past where you'll find some machines made by Dr. Robotnik (famously known as Eggman). These machines will cause the future of these worlds to be a bad one and so destroying them will make them good.
In order to switch between eras, you need to find a sign in the present time. When you run past it, it will either tell you "past" or "future". Once it's on past, you'll need to run back and forth and pick up speed. Once you go super fast, you'll be sent to either the past or future. Once you go to the past and destroy the machines, feel free to head to the future which should be "good" now and much more cheerful and peaceful.
Try Pacing Yourself
I understand that this is a Sonic game and the legendary hedgehog is all about speed (while Tails is about flying), but in Sonic CD, feel free to take your time a little bit. Yes, it's hard to do in a game when you're almost constantly moving and at great speeds, but keeping a fair pace may help.
The main reason I say is because you don't want to pass those time travel signs. It can be very easy to pass them up by accident, especially if you're a new player. Depending on the level's layout, you may be able to go back and find the sign again if you passed it up.
The important thing here is just that you don't go too far passed the sign. Go any further, then next you know you reach the end where you take on Eggman who is famously the boss on the majority of Sonic levels. But yeah, try and catch those signs and destroy those darn machines of the past.
Give Time Attack a Try
Time Attack in Sonic CD is a fun extra mode that you can play. It's pretty much what you think it is. The goal is to complete a level as quickly as possible. First, however, you need to complete levels in the campaign in order to unlock them for Time Attack.
But even after just completing Palmtree Panic, the first area, feel free to jump into Time Attack. It's not only a fun, more casual mode, but it's great for practice too. Feel free to go and work on things like timing your jumps and learning more about your surroundings.
Playing Time Attack is a fun way to sort of warm-up for your next playthrough if you choose to do such a thing. And, it's just nice if you want to get a quick run in some of the cool areas.