I used to use a Tumblr for my comic, but now I just use a wordpress/comic easel with own fancy domain name and everything right here and I love it.
While Tumblr kind of worked for me while I used it,, it was always a very boring and generic looking site. The customization options were rather limited. The plus, however is that it was very simple to use, it took me under an hour t get a comic Tumblr set up. Still, I could never stand how unprofessional it looked so I eventually switched to getting my own site and just using Wordpress.
Now for getting your own site, I don't think hiring a web designer is really necessary if you're willing to do some tinkering. I knew literally nothing about webdesign/html while making my site, and I think it came out pretty decent, and I only had to write like half a line of code. The Comic Easel plug in I used with Wordpress already gave a lot of options for customization, and I've seen some people make pretty cool sites with it. It can take a while to learn how to use Wordpress, but once you've got the hang of it, it's pretty simple.
The main con of having your own site is the cost. I think on average a domain with hosting costs around ~$50 per year, assuming you don't get insane traffic.
Overall, I prefer having my own site a lot more now that I have one, and I would wholeheartedly recommend it. Tumblr is perfectly functional, but I just don't like using it because of how samey all tumblr webcomic sites look. If you're just dipping your toes or are tight on cash, a Tumblr might be better for you.
Good luck with getting a site set up!