Hi Denis,
1. First thing first: do a DB backup!
After you go to Tools > UTF8ize you should read the first notice:
If you run the following SQL statements, you will convert all your database character sets to utf8, trying to follow Codex guides https://codex.www.ads-software.com/Converting_Database_Character_Sets
2. You can build one big SQL file that will convert your site, by copy-paste the SQL commands from Tools -> UTF8ize to a file named let’s say myutf8ize.sql
3. Run your SQL file against your database using a tool like phpMyAdmin or via the MySQL/MariaDB command line if you are hardcore like that, see https://dev.mysql.com/doc/refman/5.7/en/mysql-batch-commands.html
$ mysql
mysql> use ‘database_name’;
mysql> soruce myutf8ize.sql
Just let me know if you need more help about this. Cheers!