In agile projects every sprint timeline we have:
- new functionality to be tested (the size of this part is nearly the same every sprint)
- regression & verification of bug fixes (and this part becomes bigger from sprint to sprint)
No automation, More informal scripted tests (test cases), Less Exploratory testing
Commonly testing in agile is started when new code arrives, but in this case testing loses precious time. Ideally tester should participate in scrum meetings to have full understanding of user stories, so he could represent end user interests. This early phase gives you sketches of test cases.
Then as software arrives you check it with this informal test cases (extending them according to your new knowledge of the software). This is the first phase of testing, scripted manual testing (there should not be too much exploration) to check basic functions of software. This is how you test new functionality.
Regression & verification
More automated tests, Less scripted tests (test cases), and Exploratory testing as a separate part of testing process
As soon as core functionality becomes stable (business process or part of it could be accomplished) you should start with automation. Core business functionality should be validated with automated tests. This will save you time on regression & verification to do exploratory testing without scripted test cases but with great mind efforts. Be sure Exploratory testing is nearly useless if the basic features are not working.
P.S.: please do not mix Exploratory testing with Monkey testing. The first is about emphasizing the dominant thought process involved in unscripted testing, but second one is about random input to the tested system. Professional tester should not do monkey testing (there is an exception: randomization of automated tests, but generally this is not monkey testing)