If I were in your shoes I would look at Ikea for storage solutions. Quite reasonable and they have some really fun ideas for children's toys. I like storage cubbies and plastic bins for kids things - it is something that will be part of their early learning years. Unless you want to stop every 5 minutes to sort them you need need storage that is accessible to them and label with pictures/words for the ID - it will also make tidying up quicker and maybe a game for the 3 year old etc.
I was going to do my craft area with items from Ikea but we already have so many large pieces of nice furniture (inherited) that I just repurposed what we have (I'm in the corner of the large family room). At the moment my things are in a pie safe my great grandfather built (with plastic dividers/drawers inserted) and two drawers of a chest of drawers and I work on a mat on my Grandmother's kitchen table. It works really well when I tidy up and put it all away!!!
FWIW I never mastered the not looking like a bomb just went off. For me that was part of having kids. Good luck.