It really depends. That was one of the first rosaries I ever prayed and obviously, I didn’t really know exactly what specifics to pray which is why I went for the general intention.
I pray a rosary nearly every day, and tend to go with what is needful - recent intentions have included a speedy recovery for my FIL after a serious operation, the state of mind of my second daughter, who is struggling with faith, friends and gender identity and for God to bless a work colleague. Other times I just ask God to use my prayers as He sees best.
One thing I often do when I don’t have a pressing intention, is dedicate each decade to my immediate family - my husband and each of my four children.
Oh, and in terms of the beads themselves - when I was much younger I was a bit of a magpie about rosary beads, I loved all the really pretty ones. But once I actually started treating them as a real ‘workhorse’, praying rosaries, and the Divine Mercy chaplet daily, and carrying them with me all the time, I broke so many pretty chain rosaries it was ridiculous! I discovered wooden corded beads and haven’t looked back - I have three sets now, one by my bed, one in my bag and one I put in my back pocket all the time. I love knowing they are there when I am at work even though I can’t actively use them on the shop floor, it’s a bit like Our Lady is there to hold my hand when I reach into my pocket and feel them there.