Prettiest nature places in Vancouver?

Almost anywhere!

If you're walking/transit in and around downtown, there's the seawall, which you can use to walk clear around Stanley Park - about 8 miles/13km if you do the whole loop. Plan for a few hours or more. Stanley Park is beautiful and huge (coastal rain forest) and is criss-crossed with trails, plus you can access the aquarium, see the North Shore mountains and bridges, and quite frequently also see whales/otters/seals/Eagles along the shore. Walk from Canada Place to experience some of the urban beauty.

Queen Elizabeth Park is a bit more manicured, but has an incredible view of the city and is accessible by bus or the train on the Cambie Line. 

The Ubc Endowment Lands Ecological Reserve have plenty of easy hikes through deep coastal forests and are really interesting, and the UBC campus itself is quite pretty. There are also a number of museums there if that interests you.

For one of the most spectacular views, you can also head to the North Shore. If you have a car, head up to the Cypress lookout for a view (weather permitting) of the whole city out to the US border. You can also do some hiking on probably hundreds of miles of trails on the North Shore - be prepared though, because it's quite rugged. These are legitimate mountains. There are also downtown shuttle buses to Capilano Suspension Bridge and other sights on 'the shore' if you want to use those.

Everywhere Quinn Duffy mentioned is spot on! I would also just add Lynn Canyon Park as an alternative to Capilano Suspension Bridge as Lynn Canyon also has a suspension bridge (albeit slightly less impressive) but there's no charge to get to it and there is far more surrounding area you can access for hikes. 

Also, Bowen Island is not far from Vancouver and definitely worth a day trip. 

Head to Steveston  as well for some beautiful expansive views. I second the UBC Endowment Lands. Spanish Banks Beach

is my favourite beach, and you can walk the seawall all the way to Stanley Park! For a view, hiking trails along Mt Seymour , Cypress Mountain or in Deep Cove (Deep cove is easiest) provide spectacular views of Burrard Inlet

well the most well-known park would be stanley park.

It is a great park with a lot of trails that you can choose and it also has the bike trail by the river which is fantastic but little cold for winter.

Also there is capilano suspension bridge but capilano park(?) has a admission fee. They decorate it for christmas with lights and it is such a romantic place to go to, and also great place for kids as well

In Queen Elizabeth park, you can look down on vancouver city and during the night it presents a nice night view.

In addition, grouse mountain's gondola has stunning night view of the whole vancouver downtown and north vancouver.

Stanley Park, walk the seawall. Queen Elizabeth Park, enjoy the gardens and view from the top. Take a gondola to the top of Mt Seymour on a clear day.  Hang out at Kitsilano Beach. Go for a walk on the Ubc Endowment Lands Ecological Reserve, for example, Pacific Spirit Regional Park. If you want to go farther afield, hike the Stawamus Chief in Squamish.

Go to Stanley Park, the Capilano Suspension Bridge, Jericho Beach, or Pacific Spirit Regional Park. 

Other's have great recommendations for you so far. I would add Grouse Mountain and take the Grouse Mountain Skyride. Lovely hiking up there and great views of the city. If you have time to get out of town, take the ferry to Victoria and see Butchart Gardens Since that is a bit far, another option would be to take the ferry to Nanaimo for hiking and some lovely inns if you want to stay the night. I believe there are tours which will take you over and a bus down to Victoria as well. Have fun!