Yes, Tasche can also mean pocket.
Tasche means a pocket in a garment (trouser pocket/pants pocket, shirt pocket, etc.) and also a bag -- typically a more sturdy kind of bag such as a handbag.
Tüte is usually flimsier: for example, a plastic bag or paper bag that you might get when you go shopping.
Do an image search for the two terms to get an idea for "typical" examples of Taschen and Tüten.
I think the original Tüten were simple triangles of paper for selling loose items such as sweets; this may be why a Schultüte is called like that though those are cone-shaped. (It's a very German thing; English Wikipedia has an article on it https://en.wikipedia.org/wiki/Schult%C3%BCte but uses the German term since there's probably no established English word for it. We called it a "school cone" in our bilingual family.)
Having enjoyed the film, "Lola Rennt" (Run Lola Run), several times I thought I understood the word "Tasche." It's a grocery bag like the kind in which Manni stashed the 100,000 Deutschmarks, right?
Yet when I was grocery shopping in Munich, and asked for "eine Tasche bitte" the cashier said that wasn't the right word for it. Apparently, the right word is "die Tüte." Can someone explain the difference? I checked the movie again on YouTube, and they definitely called it a "Tasche."
yes that's correct we refer to the whole handbag as a purse, but also use "change purse" to refer to the bag type thing for coins. I also feel that I saw duolingo say Handtasche to me at some point and would agree that that would be the American equivalent to purse. Could someone confirm this? and also I want to know for certain if "Tasche" refers to both bags like plastic or paper bags as well as being the word for pocket, which I recall from previous lessons.
Not all words change in the same way when they become plurals. Some get an "e", some get an "n", and sometimes other things happen too. Duolingo has a lesson on "Plurals" which you should revise (the explanation on the lesson webpage is helpful). You can also look for other explanations online, like this one.
Sack means a clothes pocket in, I think, Switzerland and Austria (so a pocket knife could be a Sackmesser there): http://www.duden.de/rechtschreibung/Sack#Bedeutung1b
But in general, this course doesn't teach such regionalisms but focusses on German-from-Germany.
I wish I knew.
I believe there are plans to bring the grammar notes to the apps eventually as well, but I don't know when they will be there.
Meanwhile, tips and notes for each new unit have been on the website for months.
So you may wish to move from the app to using the website on a PC.
Tablets may work as well, especially in landscape orientation, but viewing the website on a smartphone screen may show you a view that acts like an app (including not giving you access to the tips and notes, unfortunately).