From your post and replies, I am not sure whether:
1) You want to monetize your website traffic by showing push ads or
2) You want to promote affiliates offers using push ads
For 2:
If you will be promoting affiliate offers, check first how the niche is doing with push notifications on forums ect..
Push Ads are hot for many niches at the moment like sweeps, nutra, dating.
Few things to consider:
Push traffic inventory on all networks are higher for Mobile vs Desktop and mobile inventory has more competition.
The creatives you will use for your ad will matter a lot and this is what can make or break a campaign.
Using a relevant prelander will definitely increase your conversion rate.
If you are promoting an offer in a competitive niche, consider using spy tools.
As for the providers, there are some more suited for beginners, like Zeropark, Megapush.
I personally use Propeller Ads, they allow for greater optimization and are considered among one of the biggest players.
I heard good things about RTX also, want to try it myself.
All the networks are getting more strict with regards to their ad review process though (which is only natural).
Their subscriber burn rate is also high so look for big networks that get new subscribers now and then.
Finally a tracker would be good to have too.