Don't be afraid of the huge lists of counters - many of them are hardly used. For generic objects if you don't know the counter you can often get away with ひとつ、ふたつ, or the 個 counters.
Also when you say 時間 for time , にち・か for days, しゅうかん for weeks, ねん for years, you're obviously getting confused: you can actually just use かん・間 for duration - you can use 時間 for hours, 日間 for days (in a row) 週間 for weeks (in a row), 年間 for years (in a row) and so on.
The special readings for days (those that use か) do need to be learnt, though.
Learn the most common counters, sure, but then just pay attention when words are used. Sometimes it's not immediately clear which counter is right (or there may be more than one possible choice), but if there's a CD collection advertised on "5枚セット" , and then there's a set of books on Amazon and it's a 5冊セット, and then there's a recipe which requires りんご (2個) and にんじん (2本), you will pick things up.