It's always a good idea to ask your instructor for advice before you buy your own poles. 

However, there are a few options where you can't go wrong. The main brands are Leki and Exel (they are both the Mercedes of the Nordic Walking poles). My personal favorite is Leki. Another - cheaper - option is Decathlon, who set up their own Nordic Walking range called 'Newfeel' and have everything from shoes to poles. 

...but also the material. The lighter the poles the better. Poles get lighter the more carbon they content. Sadly they get pricier the more carbon they contain, too. 

...but also how the strap attaches to the handle. Every brand has a different way to attach and release the strap and it's worthwhile to compare and try!

...but also whether you want fixed length poles or an adjustable poles. The answer here is quite simple. If you don't share the poles with someone else get a fixed lenght pole as they are lighter and the center of gravity is better. Also the tip of the fixed length poles is at an angle and catches the ground better. 

...but also whether you want poles with all terrain tips/smart tips. These are a combination of hard tip and rubber paw to make an easy change from one to the other.  I think this makes sense only if you walk on mixed grounds regularly. Again as with the adjustable poles the centre of gravity isn't that ideal in my opinion.

Here again it depends on where you want to walk. 

If you stay on tarmac and gravel paths normal trainers will do. Walking boots are not recommended as they tend to restrict your ankle movement.

If you also intend to walk on muddy paths or through woods I would recommend shoes with a really good grip as you don't want to slip. I found 'trail running shoes' really good.  

If you intend to walk whatever the weather it might be a good idea to get some waterproof shoes.