"No quiero que él tenga televisión."
Translation:I do not want him to have a television.
If it's an object that's usually singular, and it comes after tener (as well as a few other verbs), the indefinite article gets dropped because it's automatically implied. However if the object is further qualified with adjectives, or if you want emphasis on there being just one (and no more), then the article is included.
There are a couple more examples here near the bottom: http://spanish.about.com/od/adjectives/a/indefinite.htm (I had a better source but I can't find it at the moment)
And of course it's omitted after ser quite often, which doesn't really fall under this "rule", but is kinda related conceptually.
Edit: Found it, section 3.3, specifically 3.3.8 (but the whole of Chapter 3 is very informative about article use in general). I can't direct link to the page because Duo messes up the URL, just click on preview and navigate to the chapter: https://books.google.co.nz/books?id=YZq0AAAAQBAJ
"Tenga", a subjunctive form of "tener," is used here because it is part of a dependent clause following a independent (or main) clause expressing a wish, want or desire: "No quiero" (I do not want). Don't want what? Using the strict form of the English subjunctive, the answer to that question would be "that he have a television," or :"que tenga television."
Before someone says "Oh, in English, we would never say "that he have a television,", yes, it is generally true that following "want" we usually substitute an infinitive phrase (to have a television) for the subjunctive construct. (We do still use it, though, with other verbs => "We insist that he attend the meeting," "They insist that he sign the contract before moving in," etc. )
A program? Oh, wow, do you mean an online program like Duo or a commercial program like Rosetta Stone, Pimsleur (?), or Visual Link? (I used all three of these to some extent; for me, VL was the best of them for learning to construct basic sentences; Pimsleur is great for vocal practice, but very tedious in the early going.). None of these are exactly cheap, but fortunately, there is so much available online now, I question whether such programs are really reccesary.
So- if you're not considering a commercial system, for the time being, I suggest just to continue what you are doing here but with frequent reference to the many online resources such as Study Spanish and SpanishDict (keep a window open to the conjugation section of SD's dictionary as you go through the DL lessons.). Youtube also has some great resources. If you can afford it, consider some private tutoring from a service like iTalki or Verbling. But the main thing, I believe, is to give yourself time — expect plateaus, humps to get over, and all that, but keep plugging away at it. Few things are as exhilarating as your first real conversations with a native speaker. Vale la pena.
So if "I don't want to have a television" we would just use the infinitive 'tener' in the second clause, but when I don't want someone else to have something, that's when we need the subjunctive in spanish? I'm assuming "No quiero que él tener television" is just nonsense in spanish?
'No quiero tener una television' is a simple statement that 'I don't want to have a television.' There is only one clause, so there can't be a main clause/ subordinate clause relationship and so - yes - you do not need the 'que' to introduce a dependent subordinate clause. There isn't one.
"I do not want him to have a television?" Yes, but in other cases in the subjunctive section Duo would have accepted "I do not want that he have a television."
Such a construction uses the (admittedly seldom used) English subjunctive, which sounds odd to some, but in my opinion, can help bridge between the languages.