You are right, most of the time you should use "der/die/das" when you would use "the", "ein/eine" when you would use "a/an", and none when you would use none in English. Giving exact rules is nearly impossible (try it for "the" and "a").
Generally if you talk about something which was already mentioned, or which is common knowledge, you should use "der/die/das". If you introduce new information, you should use "ein/eine". If you are talking about unique, singular things like "Frankreich", you don't use any article at all (but strange enough it is still "das Universum/Weltall"). Sometimes you leave out the article as well when you talk about general concepts like "Freiheit".
Getting a feeling for the correct use of articles is really hard for people who speak non-European languages. Speaking fluent, natural English will be very helpful for chosing the right German article.
As you see in the comments, the situation isn't even clear for countries, so as a little bonus here a link to a list of countries with article:
Of course, if you have an official country name including something like "Republik" or "Königreich", you need an article, too. E.g. "die Bundesrepublik Deutschland" :-)