Children can only be adopted once you have a property with sufficient room for them, in most cases this will be at the sacrifice of something else such as the Alchemy Lab in Breezehome. Various beggar children can be adopted or after a certain quest necessitates a change in governorship, there are a handful available from honour hall orphanage in Riften
Marriage can be begun by travelling to Riften and speaking to the priests at the temple of Mara as I recall....after discussing with the priest you need to wear an Amulet of Mara and approach an eligible individual. A full list can be found on UESP. Some, if not all, will allow you to live with them until you have a property of your own to move them into, but I don't believe any have enough space for adoption.
To get married u need the The Amulet of Mara one way of getting it is by going to riften and talking to Maramal. The put the amulet on and potential spouses will notice the amulet and ask about marriage. To get some potential spouses you have to do quest for them before they get interested.
When you finde the one go to the temple of mara in Riften and and the the priest that you want to have a marriage.
As for adopting a child first thing you need it a house with a childrens bedroom then an option to adopt will show up when talking to a child that you can adopt.
Then you need to talk to steward in dragonsreach and get a childrens bedroom for your house.