Should I get an Apple Developer program account to develop my application or can I do it without it and then submit it obviously through an Apple Developer Progam?
There's no need to pay for a developer license to use Xcode, compile and run things in the simulator and even to install your software on your devices (macOS, tvOS, watchOS, iOS) for testing and even use as long as you're putting your personal code on your personal devices.
When you do need to pay is when you want one of any of these:
- Professional Apple Support / code level incidents from Apple Developer Technical Support
- Verified code signing certificate that let you sign code to run on other computers (whether you use the App Store or not to distribute)
- iCloud data storage services (sync, core data, cloud data, cloud preferences) for your applications.
- Presence on the various App Stores to take payments and list your applications publicly under Apple's sales presence on the official stores.
- TestFlight beta testing services where others in addition to you can test pre-release versions of your software.
Go without the paid access until you need one or more of the above services. See this link for more details - https://help.apple.com/developer-account/#/dev7a05c5ce1