I think the modifier called " military presence near the city states " count for 10 tiles around the city states , so you can prepare some roads that will allow you army to move in faster for the " dance " of tributing.
If there are two city states that are 15 or 20 tiles apart you can keep your military almost static in the middle of those.
The early tributing before turn 100 or so ( early classical) , i'd call it lucky , if you have a warrior upgrade to spearman , or if the city states gets its first units destroyed a few time , or if the city state is on flatland and no hills , or if the city states is not hostile nor militaristic , or again if the city states is not protected by anyone those are stuff that will allow for easier tributing early.
Later my personnal opinion is that it is not worth to make an army just for tributing city states, usually the AI have quite good armies so you will need to have yourself a big one in order to bully , it doesn't require much more investment to have a competitive army against the AI than just to tribute you so migh as well wage a few war with your neighbours , you can tribute empire too ^^.
Don't bully a city state that is under someone's protection if you are not ready to fight this guy, as they will ask you to stop, and it can causes a war if you reply them to take care of their business. And if you always lies to them telling you'll stop and you don't , they will denouce you and tell the world about your perfidious actions.
Do : try to bully a city states before you conquer it , it's not like you will have a bigger diplomatic penalty because you tributed them before killing them it's just free bonus.
If you plan to keep friends while still attempting at being a bully , there is a WC resolution that lowers the penalty from being aggressive towards city states.
If player 1 and 2 are at war with player 3 , I am pretty sure you can tribute the city that are protected by or allied with player 3 without consequences from player 1 and 2 and vice versa, so you might pay some AI to declare war on statecraft AI , and then get your money back with benefits allowing you to be more proactive than just waiting opportunities.
There is a time limit to prevent you from tributing the same city every turn, it is shown as a red negative modifier " -9999 you asked a tribute (very) recently " or something like that when you try to see if your green modifier showing your overall military and your close-to-the-city military are enough to balance the modifiers that come from city states resiliance, its allies , protection ect This is a bit like the system that prevent the mongol to annex all the city state in the same turn for example.
Later in the game there is " Exploitation " in the Imperialism tree that boost your ability to tribute and interact with city states and the strongest policy for that purpose would be "Tyranny" , a 3rd level tenet of autocracy that let you tribute your own allied city states further increasing your influence but nothing in Fealty or Artistery , which leads to the last point I have in mind , DON'T think you can't take statecraft while being a bully.
good luck picking on the weak and bringing injustice to the world , it's fun