In terms of Palm Cove itself, it's basically a strip of beach lined with nice hotels, high quality restaurants and cafe's/bars. The restaurants at Nu-Nu's and Alamanda have the best reputations, but others are good too. Chilli Cafe is nice for a coffee and people-watch. 2-3 days is all you'd need here, or longer if you enjoy lazing round the hotel pools.
Plenty of day trips run from Palm Cove, though you'll almost always be picked up by coach and taken to Cairns or Port Douglas from where they really start. The most obvious are trips to the Reef. For beginners a trip to Green Island(Cairns) or the Low Isles(Port Douglas) are both good starting points off tropical islands. If you're already a good snorkeller or a diver the best day trips are day boats out of Port Douglas ( Calypso is very good) while if you're a really experienced diver 4- day liveaboards to Osprey Reef from Mike Ball or Spirit of Freedom are excellent but pricey!
Another good day trip is the Skyrail (cable car) up to Kuranda in the rainforest.
Cairns Zoo ( which is actually right by Palm Cove) is only small but I'd also highly recommended it, but is unfortunately set to close by about April 2016 I think.
Having your own car, though, really opens things up.
Cairns is about a 30 minute drive, worth it for a walk along the esplanade, a dip in the swimming lagoon (no crocs or jellyfish guaranteed!) , an ice-cream or an afternoon drink at Mondoes or The Salt House. Waterbar is good for a meal on the marina.
Port Douglas is 45 mins North of Palm Cove along a stunning windy seaside drive, with loads more cafe's and bars and restaurants even better than Palm Cove.
Further north is Cape Tribulation over the Daintree Ferry, a good 2 hour drive from Palm Cove but well worth it... Spectacular rainforest and beaches, ideally staying over so it's not such a rush.
Again Atherton Tablelands is good for a day tip, or longer. Yungaburra is a nice little town to stay in with the chance to see platypus down by the river at dawn or dusk.
Mission Beach 2 hours further south from Cairns is also worth a look if you have time.
One local secret you won't find in the brochures is Behana Gorge, a spectacular waterfall and swimming hole about 25kms south of Cairns. You do have to walk for about 45mins from the car park to get there, but it's well worth it. Weekdays outside school holidays you might well be the only one there...
So how long do you need in Palm Cove??? Well I've lived here 12 years already and haven't run out of things to do yet...
Best time to visit, May to November, but anytime is good, though February or March you could get unlucky and have a week of rain... Makes for some spectacular waterfalls though!