Lightroom has the Save command like all other programs; on Windows use Control S and on Mac use Command S.
After adjusting your image in Lightroom, executing the this command saves out your Lightroom changes in one of the following ways:
1. If the original file format can contain the updated metadata (PSD, TIF, DNG, JPG) the changes will be stored within the file.
2. If the file format cannot have metadata saved into - eg native camera raw files (CRW, CR2, NEF etc) a sidecar file will be created with the .xmp extension.
Regularly saving out your Lightroom edits is a good idea, regardless of the type of file you're working with.
Capturing in raw, converting and editing DNG is the best method.
If you later edit the file in Bridge/ACR, you have to use the Done or Save commands there, too, to update the metadata edits. Then, you can use the Synchronize command in Lightroom to update the file in the LR database.
There currently is no way to automatically keep everything synchronized between programs if you're not saving out the edits.