Initially, I just draw stuff on paper (or whiteboard if that’s your preference). Just pen and paper for quick diagrams and sketches, writing down notes, etc. I have arrows flying all over the place, and notes besides the arrows, etc. (I don’t use drawing/prototyping apps for this activity since I have a tendency to get bogged down using the app itself, or getting sidetracked by other unimportant things, or I just find it too slow to jot down ideas.)
The diagram could be a series of steps needed for a task, or the diagram could be outlining the relationship between different database tables (and related/key fields) to see if it will work or makes sense.
Next step, I start creating the initial database tables and fields, and populating it with some test data (manually entered, or if I need a lot of data, I create a simple program using some Faker data. Then I can populate the database with thousands of test records in a few seconds.
Then I start testing my algorithm, writing SQL queries and manipulating the data to see if I can come up with my expected results. At this point, I may think of adding additional fields in the tables, as I fine tune or I remembered something, or nice-to-have things to keep track of… (sort order, registered Date, IP address, suspended, paid, confirmed, validated fields, etc)
Once I have a working SQL Query, I save this query so I can remember it (and not re-invent it again later). At this point, this is where it’s useful to have some system that can keep track of your notes, bugs, issues, to-do list, etc. As @honmanyau mentioned, I use JIRA app for this… running locally on my OSX.
Next, I’ll start setting up my development environment… creating the local webhost, configuring the web server, creating IDs/passwords for the SQL database, setting up my test local DNS’s (using HOSTS file, named http://projectname.mylocalmachinename.com). I’ll start setting up my LESS/SASS/JSHINT/Minimizer/etc directories and doing a few test run to make sure files are compiled and created in the correct directory. Depending on project, I can use bower, or codekit, or plugins/extensions in Visual Studio, whatever.
https://codekitapp.com/
I’m sure I’m leaving out a ton of steps in this writeup, but thing is no matter how long/careful you plan it out, it will change as you go along and start coding. You may find out the algorithm you thought out is very slow, or bogs down exponentially when you now have thousands of records of data, or just plain wrong, or you’re trying to implement a new feature and now your initial algorithm isn’t working out.
One thing’s for sure, the best laid out plan will still change and evolve as you start developing your app.
But even an incomplete/imperfect plan is much better than no plan at all. 
Good luck on your project!