Rebuilding an engine is a time and money consuming job for most people, so you want it to be done properly and that means using the right lubricants.
When putting all the parts back together, some people use thick oils (such as a 20w-50), but there also purpose made lubricants, such as Millers Competition Assembly Lube or Red Line Assembly Lube. They will provide immediate lubrication when you first start the engine, but then dissolve into the oil so there are no issues with the thicker stuff causing blockages.
Ideally, you want to use a mineral oil for the running in process as you don't want the oil to provide too much protection as the components need to wear together. Millers CRO 10w-40 is ideal for most engines, even if they wouldn't normally use a 10w-40. As you want the rings and liners to wear a little, using something that isn't the exact grade for the engine is not a problem. Often a semi-synthetic can also be fine, but I have heard of a fair few rebuilds that haven't bedded in properly when they have used a semi. There have also been plenty that are fine though.
There are different views on the running in process. I tend to suggest filling it with oil and starting the engine up. After approximately half an hour or 30 miles, drop that oil out and replace with a fresh fill as the oil that you drop out will contain a lot of metal particles. After another 200-250 miles of varied revs, drop out and replace that fill of oil, again there should be a fair amount of metal in that oil. I would then do another 400-500 miles with that fill of mineral oil and when you drain it out, there should be a minimal amount of metal in the oil. After that, refill it with the oil that you intend on running the engine on. Perhaps I am over cautious with the 3 fills of mineral oil, but to me it is sensible to make sure that it's properly run in after the time and money the rebuild has cost you.
Other people suggest running the engine in on a dyno, with varied revs and that is the sensible option for running in a car that is not road legal.
Using too good an oil (synthetic) for running in can lead to bore glazing as it protects too much when you are trying to get everything to wear together. That tends to lead to higher fuel and oil consumption, as well as reduced power. There are products that can remedy that, such as the Millers Glaze Bust.