When I start my car (2003 Lexus ES300) it idles at around 1200-1500 rpm for a good five minutes or so. After that time is up the idle will drop to 500 rpm though from what I can tell, about 750 rpm is normal for idle. Also when I first start the car I can feel quite a few vibrations, maybe something to do with the motor mounts? Also also, I can hear a tapping coming from the engine bay when idling, though this doesn't happen everytime I start it. Any help is appreciated, thank you.
Rough idle can be produced by various things:
1. A dirty engine air filter
2. A dirty or defective MAF sensor
3. A dirty or defective throttle body
4. A vacuum leak (hoses going into the engine)
5. Spark plug EOL (End of Life, cracked, covered with oil, loose, old)
6. Coil plugs (similar)
7. A clogged fuel line, fuel filter or defective fuel pump
8. A failing sensor (O2).
9. A defective throttle position sensor.
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A scan tool with enhanced diagnostics and live data would provide some clues as to which or how many of these issues would affect your idle.
Do the simple things, first, such as the air filter and the MAF and throttle body cleaning.
See Scotty's video on cleaning the MAF and throttle body: https://www.youtube.com/watch?v=lKV2uo3tlx0
As for the vibration, the motor mounts can be visually and tactically checked. See Scotty's video on that.
https://www.youtube.com/watch?v=lKV2uo3tlx0
If there's a cylinder misfire, which is possible, the scan tool will pick that up. If the air flow problems are not the issue, then it may be either a spark plug or coil issue, or an actual cylinder/piston issue. Again, begin with the cheapest and easiest strategies. And, get the car scanned, especially if the simple fixes don't resolve the major issues. (At some point, a wet and dry compression test may well be in order. Scott's got a video on that, as well).