You can create a MyNCBI account by clicking the link in the upper right hand corner of PubMed. That allows you to save your search history by using the "Create alert" link. You may set an alert to email you new search results as they're added to PubMed.
You can export references from PubMed to citation management software such as RefWorks, Zotero, Mendeley or EndNote. Some of these programs have a bookmarklet you can use to easily import one page at a time of references.
You can also export to RefWorks or other tools by choosing "Send to" - "Citation Manager." This will save a file with an .nbib extension which you can then import into your software. Please note you can only export 200 results at a time this way. If working with larger sets, you may prefer to use Ovid MEDLINE.
If you don't need to export all of your search results, you can select individual references and send them to the Clipboard. Then you can export those results from the Clipboard screen.