Any proofreading or text corrections should be made before any print house EVER sees the piece. If you are having pieces printed and finding typographical errors on them, that's an internal business issue on your part. You need to do a better job of proofing your own work before sending it to press. Print providers will not correct typos or obvious mistakes on your files. That's not their job. Their job is to reproduce exactly what you send them, mistakes and all.
When a piece does go to press....
The first line of error elimination for printing is prepress proofs. You should be getting proofs from any printer. At this stage proofing is more about visibility and color. You'll want to ensure all text and images appear on the proof. There are times where things can drop off. A PDF is generally acceptable for this proof stage because if something has shifted or dropped off the PDF will show it clearly. Sit with the proof and a copy of the artwork you sent to them and check that both pieces match in terms of placement of objects, page order, etc.
If color is critical you will also want a color proof or chromakey. Many printers nowadays will want to simply provide PDF proofs. I can't stress enough that you can not proof color via a pdf. You need a hard proof to accurately check color.
Any mistakes found on the two items above should be noted and a new proof requested. This is repeated until there are no mistakes.
Once the above two items are correct the piece moves to press.
- The next line of approval is a press check. Press checks have to be scheduled because everyone needs to be in the right place at the right time. This is where you actually go to the print house and view sheets coming off the press directly. This allows you the opportunity to review the work of the pressman directly and point out any areas where plates may be misaligned or where ink has been washed out/over saturated. You should also verify everything looks solid and there aren't any hickies on the plates themselves. (hickies are unwanted marks or spots on the plate).
Press checks are not always possible in today's "online printing" world. Most online printers will only provide PDF proofs and nothing more. In addition, some smaller printers aren't accustomed to press checks so they may balk at the notion. It's up to you to press the matter if you want.
That's about as much as you can do in-processes.
- The final stage would be delivery. As soon as something is delivered you should check the run. Pull pieces from the top, middle, and bottom of stacks. Verify color is correct and nothing has dropped off or shifted. I stress middle of the stack because often that's where a pressman will hide the less desirable pieces.
If you find an issue with the delivered pieces it is importnat to not use the pieces and contact the print provider immediately. The longer you wait, the less validity they will place on any issue you present. You should always instigate a conversation with the printer if you find an issue before doing anything with the delivered items. If you use the items, that shows your willingness to accept what was delivered.
If you are a stickler for quality or there is something which is critical to the piece, I'd urge against the use of online printers. Online printers are churn and burn print houses. They make money by gang running as many projects at once as possible without any special consideration for any single project. This means they can't tweak a color on press to ensure your particular piece looks better. Adjusting color on your job means they adjust it on any other job which is being gang run as well.
If quality is critical you'll want to use a local printer or print broker so you have immediate access to the print house and so they will treat your project as an individual.
The mark of a quality printer is not how well something looks when things run smoothly. The mark of a good printer is how well things run when there is an issue. In my experience, any quality printer will always listen and objectively look at any issue you present. If there is an error on their part they will often offer a discount or a rerun if the the issue is major. However, if the issue was present at any stage of proofing above, the onus falls upon you to point it out. If you missed something incorrect on the PDF proof and it got printed, the printer is not obligated to correct the issue without additional costs. At this point printing costs can be quite considerable and no one will want to "eat" those costs so things can quickly devolve into a finger-pointing game. Be prepared for that by showing how things were correct in proofs and incorrect in deliverables.