Absolutely start in the warehouse! I feel this will give you an edge when you go to qualify to be a driver and it will give you respect too for the part timers because you will get to see what they put up with. You will get a feel for what the company expects from you too. Many off the street hires that try to qualify don't understand the sense of urgency you need in this job.
Also if you get to load trucks i feel this helps because you will see how the trucks are laid out and supposed to be loaded. You will learn how to read the labels and how to arrange the bulk and bigger packages. When you go into driving you need to know how the packages should be loaded in the truck so you know where to look when you make a delivery. Also you need to keep your truck organized through the day and keep moving your packages forward. I feel like working preload first and loading trucks before i went into driving helped me with all these things.
If you start in the warehouse, when peak comes you can sign up to be a driver helper. So you will work your normal shift in the morning then have a few hours down time and then you meet a driver out on their route and go with them for 3 or 4 hours and deliver the packages while they arrange their truck between stops. This pays more per hour but can also help give you an idea what its like driving. Let your driver know you want to become a driver and he/she can give you tips and tell you more like how it is.
I applied tried to apply at fedex and Amazon as a delivery driver, they require commercial driving experience even if you don't have a CDL, they require work background driving with references that include name and phone number.
Your suggestion that I start off as a warehouse worker to become a delivery driver doesn't make sense.
What they require for a delivery driver is to have a clean driving record and to drive quickly to the delivery locations without getting lost or having a crash while driving a large van. Many people can't drive a large van even if they have a regular driver license.
If you can drive a large van without running over things or crashing into cars and are familiar with the roads than you should apply for a delivery position however, if you suck at driving and have never driven a white box size truck size van than you shouldn't apply for the delivery position cause you gonna end up crashing it and getting fired. For instance if you miss an exit and get lost your going to be screwed around trying to move around with a huge delivery truck trying to get to a location to either pick up the packages or deliver them. And people are usually waiting for the delivery driver esp. business locations.
UPS doesn't care if you can lift boxes, that's why they have the low paying job for those warehouse workers. Delivery driver requires being able to drive not being able to lift boxes.