SQL - Continue running all SQL statements even after error

You should use 'GO' between instructions in order to contine the execution no matter on the errors:

Alter Table Table1 ALTER COLUMN T1 varchar(MAX);
GO
Alter Table Table1 ALTER COLUMN T2 varchar(MAX);
GO
Alter Table Table1 ALTER COLUMN T3 varchar(MAX);
GO
Alter Table Table2 ALTER COLUMN R1 varchar(MAX);
GO
Alter Table Table2 ALTER COLUMN T1 varchar(MAX);
GO
Alter Table Table2 ALTER COLUMN T2 varchar(MAX);
GO
Alter Table Table2 ALTER COLUMN T3 varchar(MAX);
GO

This will give you all messages and will execute all the sentences one after each other. Those are my logs on a similar situation. As you will see, various errors are notified, and not only one:

NOTE: The catch behavior depends on the severity of the error, this link from the MSDOC explains how try_catch works https://docs.microsoft.com/en-us/sql/t-sql/language-elements/try-catch-transact-sql?view=sql-server-2017

Hope it helps :)

Use a try-catch block:

Alter Table Table1 ALTER COLUMN T1 varchar(MAX);
Alter Table Table1 ALTER COLUMN T2 varchar(MAX);
Alter Table Table1 ALTER COLUMN T3 varchar(MAX);

BEGIN TRY
     Alter Table Table2 ALTER COLUMN R1 varchar(MAX);
END TRY
BEGIN CATCH
     print 'error altering column R1 of Table2';
END CATCH;

Alter Table Table2 ALTER COLUMN T1 varchar(MAX);
Alter Table Table2 ALTER COLUMN T2 varchar(MAX);
Alter Table Table2 ALTER COLUMN T3 varchar(MAX);