"Hele" means to travel. "Ku'ai hele" is the act of shopping.
"Hele au i ke ku'ai hele," I travel for shopping.
"Ke hele nei au i ke ku'ai hele," I am travelling for shopping.
"E hele ana au i ke ku'ai hele," I will be travelling for shopping.
Substituting "hele" for "go" seems to clutter things up in translation. The principle action in this sentence is travelling. The object is shopping.
Hele au i ka hale kūʻai/I go to the store (because I get more exercise wandering around than from ordering nā mea no (things from) Amazon). I believe that Hawaiian uses present tense to express near future events, and if so, "Iʻm going shopping" equals "I go shopping" and is Hele au i ke kūʻai hele. I could be wrong, though. Edit Someone commented that the future tense can be created with E [verb] ana but thatʻs all I know.
Nothing in this course as it exists right now distinguishes present and present-continuous tenses, so you're right to treat them as the same. But apparently, according to Wikipedia, "ke [verb] nei" explicitly turns the action into the present-continuous.
My best guess at "I'm going shopping" would then be "Ke hele nei au i ke hele kūʻai" (again, literally a guess; I learned about nei about 2 minutes ago after seeing this thread).
You're right about "ke--nei"
I speak Hawaiian and I'm struggling with some of these sentences. So many of them feel either too specific (not accepting equal alternatives I think should be accepted) or they are too inclusive (having one structure cover present and future tense, when a different structure for future tense exists). And I never know which one will be which way (too specific/too loose).