The burning of wood is a huge biomass burning industry used to spin turbines, cook food, and heat the homes of people like you. Correctly burning wood is a skill that can save you a house full of smoke or any number of other problems. The first step is to make sure that the fireplace or wood burning stove is clear of any debris from a previous fire.
Scooping any excessive ash down an ash dump or cleaning out a wood burning stove is the first step, although an inch or two of ash can actually help the new fire burn more successfully. Keep any grates in the fireplace clean to prevent smoking problems. It can take ash up to an entire day to cool completely.
I recommend starting the fire using some paper and kindling and adding in seasoned wood. I do not recommend burning wood that is still green or wet because it will cause a lot of smoke. Seasoned wood burns the best. Open the damper half way, not all the way. 
This gives the fire an air source while still venting any smoke. Place the paper underneath your grate and kindling wood on top of the grate after the fire is lit. If your fireplace is prone to smoking problems or has a poor draft caused by other factors, carefully (using gloves) holding lit paper up into the flue system to warm it can help increase the chimney’s draft. Stack your seasoned logs on top of the kindling strategically to allow the kindling to burn fully. Open the damper completely once you have the fire really going.
As time goes on it may need adjustment or additional wood to be added to keep it burning. Using fireplace tools can help immensely
in getting a proper burn. A fire poker or a pair of tongs are very helpful for when this needs to happen. If a log is smothering kindling or it just needs stirred up a bit these tools are great to have. Scooping ash out of a wood burning stove can be cumbersome so purchasing a high powered ash vacuum can save you the trouble and the mess. An important thing to remember is to not leave a fire in a fireplace or a wood burning stove unattended for a very long time. A fire that is nicely contained in your firebox can wreak havoc should it start to burn up in the flue system. Paying attention to the fire the entire time it is lit is one way to ensure that if this should happen, damage is minimized.
Close the damper once the fire has completely stopped burning and smoking. Keeping the damper closed keeps the air you paid to heat from floating up through the chimney and heating the whole neighborhood. Fireproof mesh/ Screens is a great idea to place in front of the fireplace to prevent embers from hopping out and burning family members or lighting a fire. There are also fireproof rugs available for purchase to be placed in front of the firebox.