Like you, I used PS/Bridge for a long time, but then I came to the dark side: Lightroom, and I don't regret it at all. I still use PS (CS6 these days) for some work that requires layers. For example, portrait retouching is much easier in PS. However, it is not my primary tool to import, process, and print photographs.
Like you, I had placed my files in semi-meaningful directories. I must say that I didn't do an explicit conversion to TIFF, but kept them as RAW files until I had need to use them in some other application, e.g. InDesign, or process/print them with PS. Also, I was terrible about keywording. Translate that to "Keywords? Too much trouble."
When I started using LR, I realized that my "sort by directory hierarchy" wasn't nearly as transparent or useful as I thought. Further, I decided to move my photographs to another drive with better data redundancy.
I ended up creating a new LR root directory on the new drive, then I used LR's import, selecting "COPY" to suck in the files that I wanted from their original storage location to my LR catalog and LR directories. I also let LR use the default (I think) directory structure of <base>/<year>/<year-month-day> format. With LR's cataloging I find that I don't use even miss having separate, specially named directories.
The only fallout from that is that I noticed that I had some very old shots taken with a camera where the date/time was incorrectly set. Amazingly enough, I find I have some photographs that I will apparently be taking in 2099.
As I selected the photographs to import, I did some minimal import keywording. In effect, the keywording was nearly the same as the failed directory structure that I had earlier used. As time permits, I have gone back and added additional, more specific keywords to photos, but I found that even with the minimal work that I did on import that I am able to find a shot using LR's filters faster than my previous "let's find the directory" approach.
The reason I used "copy" instead of "move" was because I'm a coward. I wanted to make sure that all photographs were correctly stored on the new drive before I erased them off the old drive.
This was the approach I used. However, there will now follow several posts telling you that I'm a complete idiot. (grin). There are those that advocate for multiple catalogs...Michael Reichman, I believe, among them. There are some who still store their files in a named directory hierachy. Finally, there are some who have import directories, then move files to other directories as appropriate. In fact, there are probably more ways of using LR than there are of photographing landscapes.
I am explaining what I did and that I find it sufficient for my needs.
By the way, I see no reason to do a conversion to TIFF or PSD when using LR. I keep my RAWs...and there's another variation with some folks converting to DNG on import...and use LR to develop/print them. If necessary, I'll export as TIFF if I'm going to do some work in PS or some plug in such as Silver Effects Pro. That export can be handled seemlessly with LR.