Updated: Jan 31, 2019
Yes. Yes you should.
I'm not sure what the rule is for citizens of other countries, but with a US driver's license, you can legally rent and drive a car in Germany. However, keep in mind that some of the bordering countries require an International Driver's Permit, so if you're driving across borders make sure to get one before your trip. You can get one from your local AAA for $20.
I rented through Europcar (booked on Priceline.com) and paid only €91 for four days. I'm not sure if all Europcar locations are the same, but I picked mine up from the Landsberger Straße 170 location, which is also a BMW dealership, so ALL of their rentals are BMWs. I had no idea when I booked, by the way, it was just the cheapest option, and it was a nice surprise. There is also a tram stop literally right in front of this rental place, so it's super convenient to get to.
Driving in Germany is super easy, basically the same as back home in the US. I didn't really notice anything weird or out of the norm. If you're the type of person who wants to stop and take pictures of every pretty thing you see along the way, or if you're moving from hotel to hotel an have to carry all your stuff with you, renting a car definitely beats taking buses or trains everywhere, and it gives you a ton of freedom.
Although they do have bus tours to pretty much all the popular sights, the car will pay for itself if you have 3 or 4 people with you - the bus tickets can get expensive! Even being solo and having to bank the cost on my own felt like it was worth it. If you're solo and worried about the cost, you can also sign up for BlaBlaCar and offer rides to people who are going to the same destination.
Here's a comprehensive guide to German driving laws that I recommend you check out before driving there.