That said. Postgres, MySQL or any relational database is a good place to start for forum software. There are valid arguments to be made why a document db might be superior in some cases but honestly if I was writing a forum I'd probably look towards Postgres and if search was going to be key, I'd also likely have ElasticSearch as part of the solution. I would suggest whatever DB you chose don't chose DynamoDB for your first real attempt. DynamoDB gives experienced data engineers headaches. It isn't that Dynamo is bad, but there are very distinct tradeoffs you need to make and weigh to setup it up for your use cases. For the record I'd also avoid Cassandra for the same reason. Something like Postgres or MySQL is more forgiving when you are learning IMO. Other options could be MongoDB (people like to hate on it for issues long resolved), SimpleDB, AWS DocumentDB etc. Amazon's document DB is Mongo Compatible but uses their backend and is very easy to work with.
Just some ideas. Also you could look at some of the open source forum software out there. Might help you get ideas.
You rarely learn to select the right database in general though your whole career. Normally you're in one of two situations: you don't know of any special requirements you have and go with an SQL database you like, or you know your special requirement and know what to look for.
I wouldn't start with dynamodb, because it's rather specific and has some limitations you may not expect. If it's your first backend project, use something general-purpose and popular so you can find lots of resources around it. Most likely postgres, MySQL, or sqlite (if you don't need a server).
Maybe one of the quickest "traditional" ways is to buy a namecheap hosting plan with a domain. Then you'll be able to use PHP and MySQL and expose your site to the internet for maybe $7 a month.
The other option for free practice is to download Apache2 onto a linux box and connect to //localhost and start playing around like that.
Good luck!!