As a professional developer I use the below plugins to speed up my WordPress sites and those of my clients,
Watch these plugins in action in this video series
Asset Clean Up Plugin (free & paid)
Asset Clean Up is a plugin which combines, minifies and can selectively reduce requests on your site. Browsers, in order to load a website, need to be able to request certain files from the host of that site. The more or these requested the slower the site is to load. By reducing these with Asset Plugin we therefore speed up your sites loading time.
Once installed and activated Asset Clean Up you can use its dashboard in the left hand side of the CMS menu to make the required changes to the site.
NOTE: Even simply activating it without going through all the requests with a fine tooth comb will help speed things up.
Cloudflare Plugin (free & paid)
CloudFlare Plugin will add a Content Delivery network (CDN) functionality to your site. A CDN distributes your sites files on multiple servers around the world with the effect of allowing worldwide users of your site to access the site files from a physically closer location and hence speed up some of the load time of your website.
Once installed you can set up a free account with Cloudflare and set up your site with it by changing your name servers (on your domain name account) to be those of Cloudflare – these are given to you during the set up process.
TinyPNG (free & paid)
TinyPNG is an image compressor plugin which, when you upload images to the Media Manager to be associated with a post or page, will compress its size so that it is smaller for the user to download. The larger the images are that you upload the larger they are for your users to view your pages on the front end and hence is slower for them to load.
Once installed and activated you don’t really need to set anything up – it will just start compressing as you upload. Easy.
Bulk Image Resizer (free & paid)
Another important aspect of site speed in relation to images is the height and width dimensions of the images themselves. Uploading an image which is larger than required more or less means you are un-necessarily asking your users to download an image with which is larger than is required for the design of the website.
For example, if your design only ever displays images at 800px width no matter the screen size then there is no point in uploading an image 4000px by 2600px high. Another example is that if most of your users are on mobile (as is common these days) there is not that much point in uploading a larger image as a background image is mostly more of a hindrance than a help.
You can use Bulk Image Resizer to, in one or two quick steps, to resize all the images on your site or more selectively.
Speeding up your site in general
These plugins will help a lot with Page speed but there is more you can do which I wrote about in this other post giving a full process for analyzing and remedying your WordPress Site Speed Issues.
