The "best" service depends on a lot of variables. Urgency, weight, dimensions, destination, address type, etc. Your best bet would be to use a multi-carrier shipping software like ShipStation, ShipWorks, TrueShip, Ordoro, or ShipRush. You link the software to your USPS, FedEx, & UPS accounts and then you can compare different services and carriers for all your shipments instantly, print your shipping labels, and stop wondering what your cheapest and best option is. Some of those software companies will also grant you discounted USPS Priority rates (Commercial Base & Commercial Plus) through partnerships with Endicia and/or Stamps.com.
In general, USPS Priority Mail (Commercial rates, not Retail) is tough to beat unless you are shipping heavier parcels and/or you can get significantly discounted FedEx/UPS rates. Priority Mail will get to anywhere in the US in 2-3 days. FedEx/UPS will get your ground shipments to nearby zones in 2-3 days, 3-5 for farther destinations. And they will also charge you a residential delivery surcharge when shipping to residences with FedEx/UPS. Your cheapest FedEx/UPS rates will be SmartPost and SurePost (both comparable to DHL), which are slow (extra 1-3 days) since they are delivered to a USPS office for final delivery.
The USPS hate is overblown. I had a business that shipped over 2,000 parcels a day via USPS, FedEx, and DHL. Rarely had issues with any of the carriers. Also had/have some incredible FedEx discounts, but still did most of our volume through USPS as the average shipment weight was pretty low.
It's not this simple, but the cliff notes answer for parcels >1lb might be something like:
If Residential and less than 3-5lbs, then USPS. If more than 3-5 lbs, then check your FedEx/UPS service options.
If Commercial and less than 3 lbs, it's close. If more than 3 lbs, likely FedEx/UPS.
Urgency and destination zone add a whole extra layer of complexity. As does dimensional weight.
And USPS Flat Rate boxes are a joke. You will almost never put enough weight in a Flat Rate box to justify the expense. Flat Rate envelopes are awesome, however. You'd be shocked what you can fit in those and ship for $4.95. And the Regional Rate boxes are viable sometimes (depending on weight and zone, you can get a few free lbs if your items are small and heavy).
Use the free shipping supplies from the carriers. You can order them through your account and they will deliver them to you. If you need non-branded boxes, mailers, tape etc., check Amazon or Uline.
A multi-carrier shipping software will eliminate all of the guesswork. Otherwise, visit the carrier website and download all the pricing files for comparison. If you want some more details, feel free to PM me.