I'd say to make your own schematic and layout checklists. Like literally a checklist (whether it's a spreadsheet or paper is up to you, but make sure there's a way to mark "done".)
For schematics, make sure you include documentation. Add descriptive text wherever you want. Use the tools at your disposal—particularly ERC, and get to zero errors (as a general rule).
For layouts, the first thing I often screw up is to start with an imperfect schematic. Of course I don't realize until I'm done with hours of layout that the pots turn the wrong way or some such silliness. The goal is to minimize these problems. Prototype as much as you can on breadboards so you can prove that it works.
The second mistake is ignoring the physical world. Sometimes I'll put components just barely too close together. Or I make it difficult to assemble. Or I put components where there's not enough clearance in the case (e.g. redesigning a board in an existing device.) The most common is to forget mounting holes. I add text and drawings defining the maximum dimensions of the board; locking the location of mounting holes makes sure I know where things can and can't go. Check your hardware (screws, nuts) and make sure your holes are the right size and with enough clearance; also, should they be connected to ground or not?
Run DRC before you generate Gerber files. Be sure to understand and enter your fab's DRC rules before you start. There's not much more annoying than routing a big old nest only to find out they want 10 mil clearance from copper to pad and you did everything with 8 mils.
Also check your unrouted nets. I'll often use the autorouter when I have a few dozen sub-millimeter traces to lay where it looks like things are touching already.
The final mistake is to make bad Gerber files. I started using makefiles with my projects to create the Gerber zip file to hand off to the fab. That way, if I forget the board outline layer, I'll get an error rather than a rectangular board (as one would from, say, OSHPark). Once you're ready to send, check your Gerber files with a viewer along with your drill file. This is where the checklist comes in really handy ... it's often at the end when I'm all excited to get a board made that I skip a step and end up with boards with errors.