I cannot answer definitely, since I'm a native German (and also not a German teacher), but this is my experience:
Yes, you will be able to form basic German sentences and, more important, they will likely be understood, even if not immediately. No, you will not be able to form correct German sentences after a week unless you are a language wunderkind. But you will get better the more you try. And I heard there is some StackExchange site to assist you with learning German. Since you're already good in a Germanic language a lot of things will be easier because of the similarities, but also harder because of subtle differences.
There are a lot of things that intermediate learners and also a lot of natives get wrong, all of which could be the things you'd like to hear now. Things like prepositions or cases, or the use of special vocabulary. None of this should keep you from learning, and none of this will keep Germans from understanding you. There is no point in pointing out some "important" cases to learn, I think, though I would be interested in what others think they are.
I think more important than specific grammar rules like usage of accusative is a general understanding of the way the language works. Your college course should give this to you, and the only way I see in such an early phase is to either read simple books (no matter if specific for learners - children's books may work just as well), or use a self-study book. Note that if you're not just going to read or write German, but rather want to talk German, the pronounciation may be more important than grammar - at least for me, it's harder to understand someone who uses correct grammar, but has terrible pronounciation, than someone with terrible (or let's say basic) grammar but better pronounciation. But I think this is not really something for the first week. After the basics, of course, you'll get plenty of resources to assist you with this.
Be sure to find out whether you learned things correctly, since that is in my opinion where a lot of learners are lacking, especially if they fall for a lot of false friends (this is also true for Germans to learn English, for example). For quick questions, especially if they're not of common interest, you can also try the GL&U chat, which is most of the time (yes, that is a very relative definition - especially if you're in a very different time zone from Germany) manned by one or two German natives that will help you and tell you what you're doing wrong. You'll even find several examples for "I'm going to sleep" in there ;)
A few specific things that come to mind:
- Learn the proper use of essential words. You'd be suprised how many get words like sein or haben wrong.
- Form short sentences. German has a strong tendency to form long sentences, but they're not easy to get right (and require, by definition, more vocabulary). "I'm going to sleep" is the right direction, it's simple and an everyday situation.
- Don't start with all tenses or subjunctive forms. That's something for later, and again something even natives get wrong.
- When learning vocabulary, learn the gender for nouns, and which cases to use for verbs. This is basically just memorizing (unless you know naturally which one is right), yet it will improve the reception of your German a lot. I'd suggest properly learning any word you're looking up in a dictionary.