1) Theme problems
Next to theme problems, the most frequent problem I've encountered with rookies is the "mission impossible" syndrome. New constructors often try to cram an unrealistic number of theme entries into a puzzle or to create grids with very low word counts. I've learned something the hard way. Solvers don't give a darn how many theme squares you have. Editors, for that reason, aren't overly impressed either. A good solid theme with a lively, colorful fill will get you sales. A marginal fill in a puzzle loaded with theme squares or with a low word count will get you rejections. Assuming that you're beginning with 15x15 themed puzzles, I recommend a word count of 76 or 78. As for theme squares, if you've got 3 15-letter entries or 4 10-letter entries, that's plenty. In fact, you can get away with even less. I've sold puzzles with 2 12-letter entries and 1 15-letter entry across the center. I've also sold puzzles with 2 9-letter entries and 2 10-letter entries.
3) Bad fill.
Bad fill entries fall into 3 categories: boring, strained, and obscure. After your theme entries, the first thing that most editors will check out are your longer non-theme entries. You're not going to be able to avoid crosswordese and overused words in your shorter fill entries, but you should try to make your longer entries as colorful as possible.
Here are some examples of what I mean:
Good entries: GET A LIFE, JUMP BAIL, BIG SHOTS, OVER EASY, etc. Boring entries: RECEIVED, RESTORES, ASSAILED, etc. Strained entries: REWATERS, NOT MOODY, UNSTOLEN, etc.
I think obscure speaks for itself.
You may not be able to make all your long entries colorful, but some should be, at least if you want to try the NYT. Boring won't get you in trouble with all editors, but strained and obscure long entries will.
For the shorter entries in your puzzle, some crosswordese is usually unavoidable, but if your grid is filled with STOAS and ANOAS, and ORONO, etc., you'll probably have a hard time selling it. Even worse are very obscure entries with which most solvers will be unfamiliar. Also, foreign words should be reasonably common. You can't expect solvers to know the Portugese word for "tablecloth."
You have to play fair with the solver in both your theming and your cluing. Often rookies have the noble goal of coming up with new and exciting ways of cluing oft repeated crossword entries. That's great if you succeed, but if the result is a clue that doesn't give the solver a fighting chance, you're better off sticking with some of the tried and true clues. I recommend putting yourself in the solvers' shoes as often as possible when cluing. That way you may be able to avoid not only clues that are too clever or cute, but also clues that are too vague or too esoteric.
We've all been there. The longer you look at a questionable fourth entry needed to round out a theme or a yucky fill entry needed to salvage a grid, the better it starts to look. If your original thought was "nah," you were probably right. One good way to proceed is to check out the theme entry or fill entry with CRUCIVERB-L.
6) The "if I know it, so should everyone else" syndrome.
I can sometimes tell what line of work a person is in by some of the entries in his or her grid. Technical lingo known only to workers in a specific field should be avoided. Similarly, you should stay away from local place names that aren't nationally known. Ditto local stores and regional brand names.
Along these same lines, be careful about entries that relate to your specific hobbies and interests. The fact that you're an opera buff or a jazz buff, or a computer expert, or whatever, can blind you to the obscurity of some of your entries. If you're fluent in a foreign language, don't assume the solver is also.
I apologize for my long-windedness. I know I'm not the only constructor who's been mentoring. Maybe some others can kick in with some advice if this seems like the right place for it. If it's the wrong place, it's too late now!