Android is a great mobile OS but there is room for improvements if you know how to tune the system. It ships with some default configurations that you can change to make it faster.
First of all you need to enable the developer options. There is a trick that will allow you to enable this hidden menu. Tap on Settings and then on About phone. Scroll to Build number and tap on it seven times.
Go back to Settings and tap on the new Developer options menu. Scroll to Window animation scale. It is set on 1x by default. We want to change all the values in Window animation scale, Transition animation scale and Animator duration scale to 0.5x. This will make the animations snappier. You can see the effects of this change immediately.
In Developer options we also have the option to limit the number of processes the system will keep in background. This is very handy if you have a phone with limited RAM or a phone that you feel is not fast enough. Scroll at the end of the Developer options settings until you see a Background process limit option. Change the value to 4 processes at most. This should be enough. You can try less processes but some applications could crash after that so be aware of it.
Home screen and wallpaper
Try to avoid too many widgets. Every widget you put on your home screen will use your device RAM and CPU. They will also use more data so try to avoid to use the ones you don't really need. Don't use a live wallpaper: live wallpapers demand more system resources than standard ones.
Some Android phones manufacturers like to customize launchers. If you don't use a Nexus phone, especially if you have a Samsung phone, try a different launcher. Nova launcher, for instance, could save a lot of resources and make your phone faster. There are a lot of alternatives on the Play store so you can also try something different and pick your preferred launcher. I personally use Google launcher on my Sony Xperia Z5.
Finally install Clean Master. It's one of the most popular Android optimization apps. It will help you to clean the cache and the ram of your system. It will also hibernate apps that drain your battery if you want.