Plainly put, Germany's plans were quite vague. Hitler had a notion of a coming superpower showdown between the United States and Germany, which we can trace back to the 1920s, and which the more immediate conquests to the east were intended to fuel, as noted by Tooze:

one last great land grab in the East [that] would create the self-sufficient basis both for domestic affluence and the platform necessary to prevail in the coming superpower competition with the United States.

But it is important to emphasize that he wasn't talking purely in terms of military conquest, but rather in terms of Germany becoming the dominant superpower in the world in the more general sense. Certainly the need to go toe-to-toe with the US was clear enough, but his plans for that weren't. A large part was the naive belief that the United Kingdom would agree about the threat of the US and side with Germany, providing the obvious, necessary naval might needed to control the Atlantic. This obviously didn't come to pass, but in turn tightening relations with Japan supplanted that and now it was the Japanese Navy which in German minds would provide the necessary naval component, and in fact was a significant factor in German willingness to declare war in 1941 despite the lack of a specific treaty obligation.

But planning never went much further than that. Vague ideas of spheres of influence, and some rough sketching out of long-range air campaigns with planes that never even entered production can be found, but I think the best summary comes from the Japanese Ambassador, who reported a conversation he had with Hitler in early 1942 where the summary amounted to "How one defeats the USA, [Hitler] does not know yet". There were always more immediate concerns, be it the French, the British, or the Soviets, and direct, military challenge to the United States just never went beyond the day-dreaming stage.

This is a brief sketch though, and for a more thorough treatment, I would point to this older answer of mine which covers similar ground, as well as this one which is more focused on the background of pre-war attitudes towards the USA.

Hello everyone,

If you are a first time visitor, welcome! This thread is trending high right now and getting a lot of attention, but it is important to remember those upvotes represent interest in the question itself, and it can often take time for a good answer to be written. The mission of /r/AskHistorians is to provide users with in-depth and comprehensive responses, and our rules are intended to facilitate that purpose. We remove comments which don't follow them for reasons including unfounded speculation, shallowness, and of course, inaccuracy. Making comments asking about the removed comments simply compounds this issue. So please, before you try your hand at posting, check out the rules, as we don't want to have to warn you further.

Of course, we know that it can be frustrating to come in here from your frontpage or /r/all and see only [removed], but we thank you for your patience. If you want to be reminded to come check back later, or simply find other great content to read while you wait, this thread provides a guide to a number of ways to do so, including the RemindMeBot- Click Here to Subscribe - or our Twitter.

Finally, while we always appreciate feedback, it is unfair to the OP to further derail this thread with META conversation, so if anyone has further questions or concerns, I would ask that they be directed to modmail, or a META thread. Thank you!