Don't try to roll it out all in one shot. Just work on solving problems. Database is timing out? Add some logging there. Requests getting dropped between proxy and app servers? Add some logging there.
If you try to add logging across the entire infrastructure in one shot, you won't know what logs you actually need. And when it comes time to diagnose a problem, you probably won't be capturing the correct data.
For me, this looked like logging to a ringbuffer and then dumping that log with an associated error report when an exception occoured. was good enough for 99% of the errors i debugged, and we never actually needed a log-shipping solution. Logs were kept on disk and requested to be uploaded on demand when investigating specific issues.
it depends on what kind of startup u are in, what kind of product you ship, what kind of user base you have, what kind of solution you have. if you cobble together a set of SaaS solutions, ETL will be your integration challenge.
If you try to add logging across the entire infrastructure in one shot, you won't know what logs you actually need. And when it comes time to diagnose a problem, you probably won't be capturing the correct data.