Can I travel to another country from the US without my passport even though I have my green card
You can visit Canada using only your green card and no passport, provided you arrive via land from the United States.
I know of no other perfectly legal ways to enter another country without a passport and only a green card. Though of course the Mexico immediate land border areas do not routinely check documents, so you may be able to visit Mexican border towns as well.
Note that you can return to the United States with only your green card, if you have been absent less than one year. If you have been gone a year or more, you need a Re-entry Permit, or if you have been gone two years or more, a returning resident visa. It is still a good idea to carry your travel document, even if you don't technically need it, as you may be delayed at the border if you do not have it.
Finally, if you are stateless and have no passport from any country, you can use the Re-entry Permit as a passport-replacing document, and obtain visas to other countries to be placed in it. You can then travel to and visit any country which gives you a visa. If you have refugee/asylum status in the US, you should instead travel on the Refugee Travel Document, which lets holders travel visa-free to a few countries such as Belgium, Germany and the Netherlands.
It depends on the country you are visiting.
It is, for example, perfectly legal to travel to Germany with only a German ID card ("Personalausweis"), no passport is needed (I do that all the time). The Green Card is only relevant at the time you want to get back into the USA; and it is sufficient for that. So you can travel to Germany and back to the USA with only that ID and a GC; other countries will probably have similar regulations.
Note though that many airline personnel don't know or don't care about such fine details; typically, Lufthansa does not let you board a flight starting outside of Germany with only a German ID, and requires to see your passport. So you are back to square one.