Enough with Harbor Freight guys. HF may be ok for somethings, but not this.
Check the wear limits for the vehicles you are interested in just to get a sense for what you need. I'm not an engine builder, so I can only guess they are looing for clearances in the neighborhood of .004 (.1mm) MAX.
To get a clearance you need to subtract 2 measurements, OD and ID, so realistically, you need to split your allowable accuracy between the 2 measurements.
Good mitutoyo digital calipers are only stated good to +/-.001". I don't think that's good enough. And this is in skilled hands. To get less than .001" accuracy you need a micrometer and a good one.
What you need is a digital bore gage. I would only get a mitutoyo as they are the best, most reliable digitals out there. Swiss digitals are ok. SPI/Fowler are sometimes Swiss, most times Chinese and simply not good enough. They will give you the wrong answer. I would search ebay until you find a decent bore gage. The alternative is not good.
The alternative is you could use telescoping gages. I recommend Starrett 579(self centering over the 229). You would probably need a 3-4" so S579D. You can find these on ebay for cheap. It's a dead simple tool, so you won't get a bad one. Just make sure the faces are clean. But these add time and inaccuracy that you have to account for with your mic.
For mics I recommend the industry standard Starrett 436. These have been around for 100 years, and are readily available second hand in good shape. Only trick is, you need a standard you can trust to set them to. It (and the mic) should be clean and you need to learn how to adjust your mic to take accurate readings. I recommend getting an inexpensive mic stand- especially for beginners. Brown and Sharpe and Lufkin are also to be recommended, but these are sometimes harder to find.
I recommend you buy a few miscellaneous gage blocks (any brand will do- just as long as they are clean) and practice measuring them. You will quickly find that you don't really know how to measure stuff. This is good to do with a digital caliper as well. You'd be surprised how much your technique effects the measurement.
You can do good work with simple, high quality tools. I doubt new cheap tools will suffice for this. For measurements of +/-.001", I think you can get away with harbor freight etc.
Example- I calibrated my HF dig calipers. They are really good. Once you learn the proper technique, you can read within .001". But certain spots in the 6" range are just plain off by .0005 or more. My calipers read small by .001" at 2" but 3" and 1" are good. That's what you get with cheap measuring tools. They could be okay, or could be okay in spots. But not good enough for tight tolerances.