<p>What's proper way to connect ODBC datasources and execute some SQL statements?</p>

<p>TQuery and other BDE components can do it, but if I understood correctly they are now deprecated?</p>

You can use the ADO components:
1) Drag a TAdoConnection from the dbGo tab.
2) Right click and choose Edit connection string.
3) Click build
4) Select Microsoft OLE DB Provider for ODBC
5) Click Next
Then you will be able to select a database.
You can drag a TAdoQuery and connect it to the TAdoConnection to execute some queries.
The rest should be straight forward.