Basically Venom has all the powers Spider-Man has. So inhuman strength, wall-climbing, enhanced senses, and reflexes at the very least.
In addition, his suit is a living alien symbiot with it's own wants and desires. It grants additional powers like the suit being able to look like normal clothing, and it can shape itself to resemble weapons like axes, spears, whips, blades, claws, ect.
It retains at least limited knowledge of it's previous hosts. Eddie Brock (the host directly after Spider-Man) knew Peter Parker was Spidey, knew where he lived, knew who his family and friends were and where they lived ect. It can also develop limited counter-measures of the abilities of previous hosts. Venom never triggered Spider-Mans spider sense powers.