Is there any other way to design websites besides Photoshop? I want to quickly build wireframes / mockups of the new user interface and I want to move the UI elements and rearrange them.
There are several options for wireframes/mockups.
I can recommend you some software dedicated for that:
All these programs contains several UI elements and you can download and load more collections of them. Furthermore Axure is a really powerful tool for prototyping. Just let you know that no one of these ones are free software.
Recently also I've used other software like InDesign, which is quite interesting and with a lot of possibilities, but you need to know properly the software.
You might want to entertain the idea of using a responsive framework like Bootstrap, Foundation, Zimit, or a slew of other choices, to create site prototypes that you'll be able to preview easily in a variety of browsers and devices. It takes a little bit of getting used to, but there is much more flexibility provided by prototyping websites quickly by using an existing framework.
One of my favorite benefits other than being able to test on a variety of devices easily, is the ability to make changes quickly based on feedback I receive. As a small example imagine you have 20 screens that you've created in Photoshop and your client asks you to change a tagline that's on each screen. You'd have to open up each file individually and make the edits. Depending on how you construct your prototype using a responsive framework, you should be able to make this kind of edit in a single file saving you time.
This kind of approach of constructing prototypes is also recommended by Stephen Hay in "Responsive Design Workflow". That entire book is pretty much a guide to prototyping in this manner, although without a heavy focus on using out-of-the-box responsive frameworks. Best of luck on your project!