If you have a few distinct results, say theorems and examples, you may wish to break your paper in half to publish. You can ref to the examples paper in the main paper stating something like "examples indicating that every hypothesis in this theorem appear in []" and then adding your example preprint to the refs marked as a preprint.

It is a good idea to post all your preprints on the ArXiV: www.arxiv.org since this will get your work going and make it more justifiable to ref to a preprint. Conference organizers often scan the arxiv for interesting new results when choosing speakers.

It is fairly common for students to use material from special topics courses with their advisor in their thesis. Be sure to get the correct references for the material (which may still be in an advisor's preprint). It is even more important to be sure to include the correct reference if it isn't from your advisor's work. Consider references as part of the "justification" step of a proof. It's good to go to original sources and not just textbooks.

By the time the paper is accepted and the galley proofs need to be checked, you should turn all preprints in your bibliography into full references if they've appeared in print. You will also need to update the journal ref for your paper on the arxiv.

In your introduction, make references to important closely related papers: papers with similar results to yours and papers which have key examples related to your results. Often these papers will also give you an idea of where to try to publish as well. You can find them by searching mathscinet.

When choosing a journal check the editor list and make sure there is an editor who publishes in your area. This way your paper will immediately be sent to an appropriate referee. In fact the editor might just choose someone your refed in the first paragraph. If there is any major politics in your field, check with your advisor before submitting. You want to avoid your advisor's enemies.

When you submit your thesis for publication it is often good to use your graduate school as your affiliation. Most often the graduate school is higher rank than your current university. However, when you get your galley proofs, be sure to update your affiliation, especially if you have a tenure track job at the time. Some jobs won't count papers with the wrong affiliation towards tenure. After all, where are they getting any credit for your research?