I passed the last month building a web application and today I have finally deployed it on heroku, if I puthttp://appname.herokuapp.com it does appear and gives no problems, but if I search for it on any browser it is nowhere founded.
I think my problem is with the heroku dyno plan, I think that should change the dyno plan to free to hobby to see it online and be worldwide accessible but I would like to be sure before spending any money.
It is my first time deploy something in general and I have never used heroku, any help would be appreciated
It has nothing to do with your heroku dyno plan. If your site is available at https://appname.herokuapp.com then it has nothing to do with the dyno plans.
But I think you are trying to say that your site in not appearing in the google(or other search engines) while searching. That is because generally google does not show new site automatically that quickly(Unless you have very strong SEO optimization).
Simple solution of this is to register your site on google search console. Link to Search console for google.
Even after registration on console it takes some time depending upon your site for it to appear in google search(it took almost 1 month for my portfolio to appear in there which is also on heroku)
Other search engines have their registration methods too but mostly if you are on google search console then you can directly import that data on other search engines.
If you have any other question ask in comment.