as long as you add columns/tables to your database it will be an easy task by scripting these changes in advance in sql-files. you just execute them. maybe you have some order to execute them.
a good solution would be to make one file per table, so that all changes belonging to this table would be visible to who-ever is working on the table (its like working on a class). the same is valid for stored procedures or views.
a more difficult task (and therefore maybe tools would be good) is to step back. as long as you just added tables/columns maybe this would not be a big issue. but if you have dropped columns on an update, and now you have to undo your update, the data is not there anymore. you will need to get this data from the backup. but keep in mind, if you have more then a few tables this could be a big task, and in the normal case you should undo your update very fast!
if you could just restore the backup, then its fine in this moment. but, if you update on monday, your clients work till wednesday and then they see that some data is missing (which you just dropped out of a table) then you could not just restore the old database.
i have a model-based approach in my mind (sorry, not implemented at the moment) in which schema-changes are "modeled" (e.g. per xml) and during an update a processor (e.g. a c# program) creates all necessary "sql" and e.g. moves data to a "dropDatabase". the data can reside there, and if for some reason i need to restore some of the dropped data, i can just do it with the processor. i think over some time (years) this approach is not as bad because otherwise developers don't touch "old" tables because they don't know anymore if the table or column is really necessary. with this approach you don't risk too lot if you drop something!