To figure out the weekday of any given date you need two things: a list of numbers to remember and the way to calculate the doomsday number for any given year.
There are 3 set of numbers to remember. The first one is easy, just the order number of the weekday. Starting from Monday as 1, Tuesday as 2 and so on. Except Sunday is 0. So kinda like starting the week from Sunday and counting from zero instead of Monday and one.
The second thing is the anchor number for the century. Luckily these repeat every 400 years do you only need to remember 4 numbers and you can figure out all the centuries from them. The set is 5, 3, 2 and 0. This century's number is 2 so next one will be 0 and the previous one was 3. 1800 was 5 and then it starts again from 0 going backwards or from 5 going forward.
For the last set of numbers you also need to know how to figure out if the year is a leap year or not. This is also relatively easy, there is only one exception to the rule with it's own further exception. Leap years are always divisible by 4 so if a year is not divisible by 4 it's not a leap year. However years with round hundreds are not leap years except years years that are also divisible by 400.