A recent appellate court decision in the US has substantially increased the value of a good pre-filing search. Preparing the patent application is next. The time and cost are going to depend on the complexity of the invention and on the quality of the disclosure materials that you've put together to educate your agent or attorney about the important aspects of the invention. You should expect to work through several drafts of the application, with major changes between drafts. Complicated cases cost more.
Drawings are almost always part of an application.
Filing informal sketches can speed up the process of preparing the application and can defer the patent drafters fee, but it does put the application at risk if the drawing fails to show an important feature of the invention. In the US, some applications are picked up for action by a patent examiner within 4-5 months of filing. Don't count on that. Expect an 8 - 12 months delay for most inventions. Eventually a patent examiner will read your application and compare it to what he or she thinks are the closest references from the prior art.