There are a lot of podcast hosts and any of them will work. My recommended podcast host for new podcasters is Buzzsprout [affiliate link]. They are my favorite for new podcasters because their platform is clean and easy to use, and their customer service is outstanding. If you run into a problem, they will be happy to help you. As a bonus to SPI readers, if you purchase through my affiliate link, they will give you 33% more upload time at no extra cost.
To submit your show to the iTunes directory, go here: https://podcastsconnect.apple.com/.
The Smart Podcast Player is now called Fusebox.