In the case of the sentence you posted, you want an em dash, not a hyphen. It's commonly represented by two hyphens like this -- and a lot of word processors (including Microsoft Word) will convert it into a dash for you. You do NOT need spaces on either side of the dash.

A hyphen is not appropriate for this usage.

An em dash can be used in the place of commas or parentheses. For example, you could also write that sentence:

I kept my beg in my lap. It kept all my belongings including two pairs of clothes (not including the ones I was wearing) and my wallet with my I.D. and the money I had made previously.

or

I kept my beg in my lap. It kept all my belongings including two pairs of clothes, not including the ones I was wearing, and my wallet with my I.D. and the money I had made previously.

All three are technically grammatically correct, so it comes to what punctuation is more effective stylistically. Each punctuation mark makes the sentence come across just a bit differently. In this case, I think you'd be right to go with the em dash! But be careful with the dash. A lot of writers will start to use it as a crutch and really end up overusing it. Be mindful and open to changing things around so you can get a nice variety in sentence structures and convey what you're writing about as best as possible! I find I write my first drafts with a lot of dashes when I'm writing quickly. When I'm editing and moving things around, most of the dashes get cut.

Here's a link with more info!

http://www.grammarbook.com/punctuation/dashes.asp