Yes, the delta of a put is always negative but you have two choices with options - you can buy them or you can sell them. If you buy them then you own negative delta. If you sell them then you "own" positive delta.
Owning positive delta may seem a bit confusing since it's a short put. That short put represents the obligation to buy the stock and the delta of long stock is positive. Is your head starting to spin yet?
To be more specific, if you are long the put (negative delta) the simplest delta hedge would be to buy stock. Conversely, if you were short the put, shorting stock would be the delta hedge.
You can delta hedge in a variety of other ways. For example, in a Reverse Ratio Spread (aka a Backspread), you would sell an option and buy more of them at an OTM strike.
You can also buy or sell calls to hedge puts and vice versa but these are more sophisticated strategies that have a variety of other risks and should not be attempted unless under the supervision of an adult :->)
After you understand this well, if you attempt any of this, make sure that you are trading at a deep discount broker that charges less than $1 per option contract. Using a traditional $4.95 per trade (perhaps plus a per contract fee as well) will kill you.