The first step to making sure that your apps are safe is to download security software. You have it on your computer, right? Your phone needs it too. The most basic ones will scan your apps for anything that looks suspicious, but you can also find ones that enable you to locate your phone or even wipe its data remotely if it gets lost or stolen, or perform data backups. You might not be able to find one that does everything you want for free, but most of them aren't more than $40 per year -- a small price to pay when you think of the damage that someone could cause with, say, the credit card number stored in your Amazon account.
OK, so how do you know that the security app itself is safe? Do some research. I chose my security app by going online and reading reviews and articles about the best ones for my Android phone. All of the download platforms also have customer reviews, so that's a good place to go when you've narrowed down your choices. Just take the reviews with a grain of salt -- if every single one of them is glowing and uses similar language, they might be fake. Some unscrupulous developers employ people to leave fake positive reviews to boost their sales.
Also look specifically at the app's developer. Many of them will have websites. You can also see which other apps they've developed. Major publishers of the most popular games, for example, aren't going to put out malicious apps. It's also safest to stick with the most well-known platforms, like the ones associated with your phone's operating system. They have systems in place to evaluate apps before they go up for sale, as well as quickly identify and remove malicious ones.
When you have chosen an app -- no matter what kind it is -- look at the permissions that it requires before you download. Some apps will request permission to see your location, information about the phone calls you make, or the ability to read and write to your SD card. Carefully consider whether the app actually needs to do whatever it's asking. An app that helps you locate restaurants nearby would need to know your location, but should an app that helps you create lists? I say "request" but in reality, the app is telling you what it will do and it's up to you to either take it or leave it. Check the sidebar for some of the most common permissions. If you follow these basic steps, you should be able to avoid the most malicious apps and make sure the ones you already have are safe.