Travelling to multiple countries in Schengen region [duplicate]

You need to apply for a Schengen visa. As @Willeke mentioned, since the duration of stay in each country is the same you need to apply for the country of first entry, that's Hungary.

According to their website Tennessee is covered by the Chicago office. I couldn't find anything on their website to suggest they accept mail-in applications, furthermore they need to collect biometric data so it seems you'll have to travel there personally.

You should apply for visa at the consulate of the country where you enter first. If your first port of entry is Hungary, then you should apply at the Hungarian consulate.

http://konzuliszolgalat.kormany.hu/how-to-apply-for-visa

I lived in Hungary for 5 years, traveled to Vienna and Prague like you. I suggest you plan your route like this.

Budapest -> Vienna -> Prague

Bus route very cheap i suggest orange ways from Budapest. Trains may be bit expensive but there is a direct train from Vienna to Prague.

Prague is very near to Berlin (takes 4h by train), i suggest you should squeeze some time to see Berlin amazing city.

I personally think 1 week is too much in each city, but 5 days would be more than enough.