A drawing style is a very individual thing. For starters, I suggest looking at the work of your favorite cartoonists. You can emulate certain characteristics you like: maybe the way hair is drawn or feet or trees, whatever. Some of the most popular cartoonists around have obvious influences from other cartoonists. Draw different characters from different strips. The next step is simply to draw....a lot. Drawing styles don't usually hatch all at once, they evolve over time. So, just find excuses to draw. Taking drawing classes isn't a bad idea either.
When you look at the comics, try to notice how a joke is told, for example: the timing; the set-up; what is said and what isn't said. Sometimes it's a fine line between making a joke too obvious or making it too hard to understand.
It's usually a good idea not to flood the frame with too many words.
One thing I must stress is write about what you know. Don't try to center your strip around an operating room if you have no experience there. It won't ring true. Make sure your printing is legible and large enough so it can be read when the comic is reduced.
If you have repeating characters, develop a different personality for each of them. Actually write down a paragraph or two about each character, what they like, etc. Also draw your characters with different expressions and from different angles. And don't have too many characters to start out...3 or 4 is plenty. Have one main charater and use him/her in EVERY strip to start. That way, the reader will realize who the central character is and slowly get to know her/him. If you want, you can always introduce more characters later. One you have defined your characters, make a "character sketch." This is a single piece of paper that has a drawing of the main characters and a very brief description of each. It's a good idea to develop some friendly friction between characters.
PUTTING IT ALL TOGETHER
Once you have your concept down, draw a big batch. Make sure you can keep it up. Try to do 10, then 30, then 50. Show family and friends for feedback. Send 10-20 samples to local weekly papers, magazines, newletters, where ever you can. (Never send originals, always copies.) If you can get published, it will help you see what your work looks like reduced and printed. It may help you get used to deadlines, too. A book called "Artist's Market" has listings of magazines, card companies, newletters who are looking for cartoons.
When you think you are ready, submit 30 of your best cartoons (along with the character sketch, a brief bio and a cover letter) to a syndicate (the fine print you see along the borders of comics). They distribute 99% of the cartoons on the comics page. Here is a list of the major syndicates. Address to COMICS EDITOR.
235 E. 45th St., New York, NY 10017
4520 Main St., Kansas City, MO 64111
435 N. Michigan, Ste 1500, Chicago, IL 60611
WASHINGTON POST WRITERS GROUP
1150 15th St. NW, Washington, DC 20071
5777 W. Century Blvd., Ste. 700, L.A., CA 90045
Be prepared for rejection. It's one of the hardest things about this business, but much of it is inevitable. If you believe in what you are doing, keep it up. If you think you may be on the wrong track, try another concept. Persistence is right up there with luck, timing and talent.
(If I think of anything else, I'll add it later.)