Loud clicking noise when it turns on and as I drive it gets lower but still rumbling noise continues

kind of reaching on a limb here because noise issue are always hard to determine especially with out determining at least a area of origin. Just a few things i have experienced in my time I have came across similar issues being caused by a possible pulley or bearing issue from either an external pulley/bearing to drive belt or event timing belt depending on what engine is in this vehicle. reasoning behind this is as bearing cools and drys it begins to seize causing rumbling sound wile cold tell warm up also slowing engine speed causing some irratic rpm however like said reaching out on a limb here.may have Idle control issues as well being cause. a few questions does this happen only wile long period of time of sitting then start or at all starts and any dash lights coming on wile doing so Engine light Oil pressure light???

Usually in the a.m when I turn it on to go to work or anytime I first tun it on but no check engine lite turns on or overheats I was thinking it might be the belt tho? Do u recommend I start there ?

if you think it's the belt spray water on it an dif the sound goes away replace the belt as a kit meaning belt and tensioner. your cam has a chain. if it still persists have a local pro check it out as it could be a follower for the cam or a loose chain/tensioner problem.