My Samsung S5 will not Power on. Although I can still transfer Music/Video/Pictures to and from it, and it is able to charge when I plug it into my USB. How can I fix this problem, and get it to turn on without plugging it into my computer?

It won't be difficult- MobiKin Assistant for Android (Win & Mac) can help you to achieve this goal easily.
It is a third-party desktop application, designed for Android users to manage the files like videos, pictures, songs, contacts, text messages, call logs and other files on the device.
Step 1. Connect your Android phone to the computer and enable USB debugging if you haven't done this before. For different Android operating systems, the steps are as follows.

Step 2. Then tick "USB debugging opened" and click "Next" to start to install USB driver on your computer.
Step 3. Now you can click the "Videos" folder in the left panel to locate its contents to the program. Then you need to tick the videos you like and hit the "Export" button to sync the videos to computer.
It's very easy to use.Try it! Solution to Transfer Videos between Android Phone and Computer

To transfer pictures in Samsung Galaxy S5 to your computer, maybe you can ask help from Samsung data manager tool. First connect your Samsung Galaxy S5 to computer via usb cable, open the tool. All pictures will be scanned out and listed on program window. Then we can choose them to transfer to computer. How to Transfer Photos between Samsung and Computer

For data transfer between two phones, you can use professional data transfer tool, MobiKin Transfer for Mobile . Some professional transfer programs allow you to share data between android phones,
How to Transfer Contacts from Android to Android for Windows Mac

through a computer or sd card

Plug your old phone into your computer. Copy the pictures to your computer. Unplug.
Plug new phone into computer. copy pictures to new phone.

done.

If you want to transfer pictures from your Samsung phone to you PC, you can have a try the Vibosoft Android Mobile Manager; this program can transfer your pictures, music, SMS and so on from your Samsung phone to your computer. Android Desktop Manager Manage Android Data on Windows PC

Install LG PC suite in the computer......connect the phone via cable supplied (CA101 - USB2). PC Suite runs automatically else run PC Suit in PC......

Try Android Phone Manager.
Here are some related articles you can follow:
How to Transfer Samsung Galaxy S7 Photos / Videos to Computer
How to Transfer Samsung Galaxy S3/S4/S5/S6/S7 Pictures to PC
Jihosoft Android Manager
How to Transfer Samsung Galaxy S7 Photos Videos to Computer
How to Transfer Samsung Galaxy S3 S4 S5 S6 S7 Pictures to PC

To transfer contacts from iPhone to Samsung, I suggest you try Jihosoft Phone Transfer. I just switched from an iPhone 4S to a new Samsung Galaxy S6, and this phone to phone data transfer tool worked well to transfer all my contacts, text messages as well as photos, videos, music, etc. from my old iPhone to the new Samsung Galaxy phone. You can download it to have a try. I found it from the video below:

Hi,
All I have to do with mine is plug the phone into my computer with the USB cable... I can transfer everything and charge it all at the same time...

heatman101