How many miles or kilometres does the Earth travel in one orbit of the sun? Is it constant?

Earth's orbit around the sun is 924,375,700 km or 574,380,400 miles (figures from NASA).
You may consider it constant, keeping in mind that:
- it varies (insignificantly) over the years due to interactions with other planets
- it increases due to tidal effects by about 6 micrometer per year
- it increases due to loss of sun-mass (nuclear activity and solar wind) by about 10 to 20 cm per year (figures vary depending on the source).
This means that the distance sun-earth increases by about 1.5 to 3 cm (roughly, one inch) per year. That is 40 years for 1 meter, or 100,000 km over the age of the earth - or 0.6% of the actual distance.