Most servers are fast enough and bandwidth not generally an issue. There are many things you can do to speed up your responses. Most of which are related to the performance of the DNS servers and the host more than the computer itself.
As far as images are concerned, optimize the images. I would not move them offline since there are new issues that arise as a result- especially running a store front. Use PhotoShop or some kind of code module to properly size and optimize images. I do not know what your configuration is, but there may be a plug-in for your CMS, blog, or store front. I would look at these. Otherwise, I would recommend a little tool that I absolutely love and can work offline once set. It is ThumbNailer from SmallerAnimals. You can find it here: http://www.smalleranimals.com/thumb.htm It is really affordable and can be a set it and forget it option if you want. I have used ThumbNailer for many years and it is amazing! Especially when sizing and optimizing images in bulk. I don't think there is a better option. In fact, I am not sure any other tool optimizes images better. Really. These guys are geniuses.
I second the recommendation of using Google Page Insights. But I recommend another option first. If you can hit your site via a browser, some browsers have tools that help to identify load times. For example Chrome has Developer Tools that can identify bottle necks. This is what I do from time to time to check my performance and how top optimize pages and server speed.
Check where your bottle necks are and solve them first, Then follow the advice given by @Martijn who makes excellent points. Enabling compression and caching certainly helps and I highly recommend them! Just make sure you do not have other problems that you are not aware of before spending time running in other directions.