Harry
Every manager has different preferences, but here's what I'd recommend.
Tyres:
I'd go for Goodweek, Bridgerock, or Michigan. The Big 4 stats (Acceleration, Braking, Handling, and Downforce) are very crucial to your car's speed, and these 3 suppliers cater to that. I rarely see Komehome and Donelots used, but they can be useful if your other stats are already maxed out or you're just severely lacking in Tire Economy. It depends on what your car needs.
Fuel:
This one is a little more straight forward in my opinion. I strongly suggest using Elves or Beep. These give you more acceleration points, while only subtracting very few design points in fuel economy, which is widely considered less important than acceleration. Thus, the other 3 suppliers (Chavon, Seashells, and Subtotal) aren't useful most of the time since they take away acceleration, or just don't give you enough fuel points to really make a difference at all.
Engines:
I've found that choosing engine suppliers isn't as crystal clear. I would discourage you from using Rednote or Wonder since, as I established earlier, suppliers that give you fuel economy and take away acceleration can be a step in the wrong direction (this can vary based on your car design, sometimes a supplier that gives fuel economy can be beneficial if you don't have much already). However, for the majority of the time, I'd recommend you use Tifosi, Murk, or Toymotor. As for which of those 3 to use, look at your car's stats and determine it based on that. But those 3 are generally a good idea.
Additionally, you can also use another manager's engines. At level 20, managers have the ability to start manufacturing their own engines that other players can use as a supplier instead of Murk, Rednote, etc. (details can be found here: https://igpmanager.com/forum-thread/52890). These engines can be very beneficial to use, but I believe you have to be level 10 to use one (don't quote me on that, I could completely be remembering that wrong).
And that's pretty much all I have to say about it. I'd love to know other managers' thoughts on this. If you have any more questions about how to design your car, this guide is super useful: https://igpmanager.com/forum-thread/45546. I hope my explanation helped!
You can use someone else’s engine at every level, there’s no limit in using it (afaik). But very well written, Harry, good job