There are two separate task here: restore and recover database and perform upgrade.
Let's start with the first one.
Let's pretend you have instance of 10G installed.
First of all you have to create an instance and start it with nomount mode. To do it you should put parameter file initFFH.ora in correct or default location, create password file so you can connect as sysdba and run SQLPLUS
sqlplus / as sysdba
SQL> startup nomount;
Next, put copy of your controlfile to location defined in CONTROL_FILES parameter in initFFH.ora and mount your database
SQL> alter database mount;
Now it's time to restore database. If your *.dbf files are databafile copies just copy it to locations stored in your controlfile. If your dbf files are RMAN backups you need RMAN command to restore database, however based on names looks like they are datafile copies.
After you copy datafiles try to recover the database by applying archive logs. Put your *.arc files to location the Oracle can reach (defined in LOG_ARCHIVE_DEST_n) and issue
SQL> recover database until cancel;
Oracle should apply archived logs until it failed to apply the next one arch_1_56554.arc. In this moment you have to cancel application and try to open your database.
SQL> ALTER DATABASE OPEN RESETLOGS;
So here are steps you should complete
- startup nomount
- mount
- restore datafiles
- recover
- open
Second problem is upgrade. Unfortunately
https://asktom.oracle.com/ords/asktom.search?tag=migration-to-18c-from-version-10g
you cannot directly upgrade from 10G to 18 so you have to do it twice. For example upgrade to version 11 or 12 and upgrade to 18
Edit. Regarding upgrade
There are a lots of good articles about upgrade like https://oracle-base.com/articles/18c/upgrading-to-18c
In the nutshell
install intermittent version (11 or 12) in new $ORACLE_HOME
start old instance with old database as usual
Run preupgrade script from new version. This script won't update but review your installation and print it's findings
recompile invalid objects via utlrp.sql
Next step is starting new instance with old database using special command
SQL>startup upgrade
Keyword upgrade
means ignore errors caused by version inconsistency.
Actual upgrade is done by running upgrade script like catupgrd.sql
or the other scripts mentioned in Oracle upgrade guide for your version.
After the script completed it's work restart the instance in normal mode
Check alert.log for errors
recompile invalid objects via utlrp.sql
Repeat the process with target instance 18