Instructor at University of Washington here - we were one of the early closures, so I feel like we're starting to get the hang of it. Here's what I'm using:
Zoom: Zoom is similar to Skype, with better support for many-participant calling and additional features. It has a built-in whiteboard you can write or type on, mechanics for allowing students to "raise hands", and a "breakout room" option that allows you to split the class into small groups. The free version only allows 40-minute sessions, as I recall, which is a little too short for a class (though I would imagine you could do a 40 minute lecture followed by a 10-20 minute mini-assignment); the Pro version allows sessions of any length. Some schools are able to provide Pro accounts to faculty, but I can't imagine most are -- I also work at a local community college, and they don't have enough Zoom licenses for the faculty.
CanvasCanvas and Panopto: Canvas was already an essential component of all my classes, but it's now extra useful. It links into Panopto easily, which is a platform for recording lectures -- if you have a way to write on a screen, it can record that, or you can use your own physical whiteboard. Or even just write on paper and hold it up, I guess. Canvas also has a functionality called "Conferences", which (based on my cursory inspection) seems to function a lot like Zoom.
Exams: I plan to administer exams through Canvas, which handles timing for me. That does have the disadvantage that I have no way to ensure that they don't make use of the Internet or similar resources, or talk to each other during the exam. To ameliorate that, I'm adjusting my exam questions to lean more conceptual than technical -- utilities like Symbolab can calculate the roots of a polynomial, for example, but they can't write a few sentences explaining the interpretation of the roots in context. I'm also planning to have the exams open for only a fixed window -- for a two-hour exam, I plan to have the Canvas exam available for a total of two and a half hours, reducing the risk that students might complete the exam and then communicate the questions or their answers to other students. I also will be requiring students to submit their work (the process they used to get the answer) as image files or PDFs after the exam.
Teamwork: The biggest advice I have here is stay in touch with other faculty at your institution. This is a confusing time for the students, and confusion does not help with learning. Anything you can do to maintain consistency will help -- if all of the faculty are using similar techniques for online teaching, that's likely to be more helpful to the students than if every instructor digs up their own alchemy of random online tools, no matter how effective those tools might be.