Saving Custom Fields Defined in BuddyPress’ register.php

Without hacking core files your only option is to use user_register hook to save the custom fields and usually custom fields should be stored in the corresponding object meta table, meaning that if its user meta then store it in the usermeta table and if its post meta then store it in the postmeta table.

As for redirecting after the submit, that really depends on how your form works, if its the regular form then you can simply add an hidden field with the redirect value, something like:

<input type="hidden" name="redirect_to" value="http://www.domain.com/redirect_to_here"/>

but if the form is custom and not really using the built in registration form then take a look at wp_redirect() function.