The DMG describes the various ways to identify a Magic Item in the section "Identifying a Magic Item", on p. 136. We already have the TL;DR answer from A Very Large Bear, but I'm the "completionist" type. (And, to be fair, the question asks "All the possible ways to idenfity a magic item")
The identify spell is the fastest way to reveal an item's properties.
Alternatively, a character can focus on one magic item during a short rest, while being in physical contact with the item. At the end of the rest, the
character learns the item's properties, as well as how
to use them. Potions are an exception; a little taste is
enough to tell the taster what the potion does.
Wearing or experimenting with an item can also offer
hints about its properties. For example, if a character
puts on a ring of jumping, you could say, "Your steps feel
strangely springy." Perhaps the character then jumps
up and down to see what happens. You then say the
character jumps unexpectedly high.
Note that knowing that an item is a "ring of jumping" might not mean anything for the character, unless he has knowledge in magic items - simply knowing it makes him jump higher could be enough.
The item says a lot about itself
Sometimes a magic item carries a clue to its
properties. The command word to activate a ring might
be etched in tiny letters inside it, or a feathered design
might suggest that it's a ring of feather falling.
Anything else your DM wants, or, if you are the DM, whatever you want. In particular, an Arcana check along experimentation or if the item is characteristic is a common (from my experience) house rule for acquiring full knowledge of the item. I have used Arcana checks to identify items; experimentation and identifying important features of the item before making the check would give advantage on the roll. Note that this usually makes the identify spell a lot less valuable, so, as a DM, make sure your Wizard (the character who probably has the spell) who spent 100 gp on a pearl to cast it is fine with house-ruling it this way.
Variant Rule: Making it harder
Your group might be right if they are following the rule for harder identification of magic items, also described in the same section.
If you prefer magic items to have a greater mystique,
consider removing the ability to identify the properties
of a magic item during a short rest, and require the
identify spell, experimentation, or both to reveal what a
magic item does.