After about 300 miles of owning the integra from a dealership, it seems the oil is a bit too low. I checked it after after through a highway coming back home. I noticed some smoke that was barely visible.

I was wondering what kind of engine oil would be suitable and what how much would be a good start to put in the car.

 

Did you check it before leaving the dealership? It could have been low when you left there, or it could be consuming oil. You need to figure out which.

The owner's manual and service manual tell you. If you don't have an owner's manual, tell the dealership they should provide you one. They both note that 5W30 viscosity is preferred, although you can also use 10W30 if it doesn't go below 20 degrees F outside temperature. You can use synthetic oil or conventional oil; synthetic is more resistant to breakdown when hot and flows better when cold, but conventional oil is less expensive. Your choice.

Enough so that the oil level is at the top dot on the dipstick. The distance between the two dots is one quart. So, if the oil level is halfway between the two dots, put in half a quart to make it reach the top dot.

 

The dealeership said they replaced they did an oil change and passed it for state inspection. i checked the oil after buying the car and there was a good amount of oil. Thanks for the info and i'll be sure to check the manual.

 

How low is it on the dipstick? If you only drove it 300 miles and it is already low, you may have some issues.

Did you notice any oil on the garage floor or wherever you park?

 

white smoke where? out of the exhaust or under the hood? you shouldnt be loosing oil that fast, check the color of your coolant and the color of the oil in the engine, make sure everything is kosher.