There really are no waterfalls in the Reykjaík area unless you count the small falls in Elliðaá. It's easy to get to and it's a green area in the city. it's very beautiful there, so I suggest you go there if you have the time. Also, the Open Air Museum is in that area so you could make a nice day out of it.
Other falls are Glymur, which is about two hours drive and a hike, but totally worth it. It's amazing to look at and the hike is pretty cool too. Then there is Gullfoss, which you can easily reach either in a tour or by simply renting a car and heading out yourself.You could also drive further because there is a lot of waterfalls there, and on the south coast, such as Seljalandsfoss and Skógafoss, and then Faxi, Gljúfrabúi and others in Þjórsá river.
Basically, it's all a question about how willing you are to drive and how far you think it ok to drive. If you just want to see waterfalls then it's likely quicker to rent a car than going on a tour because you will have more time to skip other locations. However, tours are pretty awesome because then you can visit placed you didn't know about and don't need to stress about not finding the way and stuff.
In general, I'm just going to recommend that you start with a Golden Circle tour for Gullfoss and then do a hike to Glymur for the sheer awesomeness of it. You can pretty much do the whole nature thing there, and also away from all the tourists. Hvalfjörður is not a super busy place so you'll be fine there if you want peace and quiet near the waterfall.
But yeah: tour to Gullfoss, hike to Glymur. Good shoes and good jacket and have some awesome fun!