Hi and Welcome. My Name is ***** ***** I look forward to assisting you today with your question.
The 4S has 5 year old hardware and you will experience slow downs on latest iOS version. You can't completely increase the speed as it an hardware limitation, but there are some ways to improve the performance a little.
You can do the following steps to speed up the phone. Remember, you will not see a significant boost in performance as the hardware is very old.
Bring the multitask window by double clicking the home button and remove all open inactive/unused applications by swiping them upwards.
Go to Settings > General > Background App Refresh and turn off Background App Refresh for the applications you don't want. You can even turn off the Background App Refresh completely which will also save battery life. This will stop the applications from running in background.
If it is very slow, then soft reset it (you can do this everyday or 2). To do that, press and hold down the Power button and Home button at the same time until it shows the Apple logo. When you see the Apple logo, release the buttons and let it boot. This will refresh the iOS (smiliar to a force restart of PC)
You can also turn off the automatic downloads of your iTunes stuffs. Go to Settings > iTunes & App Store and turn off automatic downloads. When you do this, you have to download them manually.
There are the main ways to speed up the iPhone.
I'd recommend to upgrade your iPhone 4S to latest model. If you don't like large sized iPhone 6 or 6S, then you can upgrade to iPhone 5S. 5S is a good model with a decent hardware and performance. Also, Apple is going to release another 4 inch iPhone, this March 15th. You should consider upgrading to that model and it is going to have the current generation hardware which can be used for the next 3 or 4 years without any issues.
If you have any questions or would like any additional information, please don't hesitate to ask.