I'd read Secrets of Consulting: http://geraldmweinberg.com/Site/Consulting_Secrets.html
I'd drop into as many tech meetups as you can and ask folks genuine questions about their problems. See if you can help them (for free). Think of resources or blogs that you can pass on to them, then do that. If you don't find anything to help with the problems, write a blog post about it and then send it to them. In general, be helpful.
I'd also focus a bit more than "website or cloud problems". That's pretty broad. The narrower your niche, the more people will think of you as a go-to for that problem. Again, talking to folks at meetups will be helpful.
Finally, you might want to think about using online resources to find problems (finding what people complain about online and then writing about it so you develop inbound interest). Amy Hoy does a great job of outlining these in the Sales Safari talk. Here's a free version: https://stackingthebricks.com/video-sales-safari-in-action/ but she also has a $99 workshop: https://shop.stackingthebricks.com/sales-safari-101
If you like 100% focusing on programming, I wouldn't recommend consulting.
Consulting is a great way to get a broad range of exposure to every part of running a business. If you want to handle client relationships, planning, dealing with finances, payment issues, and the like, you'll like consulting. It's a very "heads up" whiteboarding-with-the-client, very social kind of work. Maybe that suits you, maybe it doesn't.
Often the problems clients face are at the intersection of organizational and technical problems, not purely technical ones.
(I wrote about my 8 years consulting here https://softwaredoug.com/blog/2020/12/22/hack-your-career-wi...)
This community seems pretty experienced and serious. I lurk there even though I don’t freelance but you might pick up good tips there. https://news.ycombinator.com/item?id=37419830
If you are still at your university, there are probably others trying to launch startups. I would sniff around and reach out to them. However, you still should work in the industry for some time first. You need to build up experience working on real projects. In consulting, no one will care much if you have a PhD, what they will care is if you have experience doing what they need. While you work in the industry, you will also build up a network. Starting a consulting company is one of the easiest things to do, but maintaining one is one of the hardest.
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Fresh graduates can program a computer but they still take a few _years_ to learn the ropes of working in the industry - all the stuff that a University can’t teach.
I feel you’re vastly underestimating the scope of what is involved with the “cloud”.