Just to clarify, you are staying:

3 nights in Prague (6/1, 6/2, and 6/3)
3 nights in Vienna (6/4, 6/5, 6/6)
1 night Veszprem (6/7)
2 nights Budapest (6/8, 6/9) and
4 nights Barcelona (6/10--6/13)

Flying home on June 14. Correct?

yup thats right...

Unfortunately you are counting days 3 times throughout. You are counting one day as a day in Prague, as the day you travel form Prague to Vienna AND as a day in Vienna. So you really have fewer days in each place than you think. Frankly I would bag Barcelona, which is an outlier and give Vienna, Budapest and the countryside the time they deserve.

nytraveler is correct. You'll be in Prague and Vienna both on June 4 and in Budapest and Barcelona both on June 10. I hope you have the correct nights for your lodging.

June 1 - arrive Prague
June 2 - Prague
June 3 - Prague
June 4 - travel to Vienna?
June 5 - Vienna
June 6 - Vienna ?
June 7 - travel to Hungary
June 8 - Budapest
June 9 - Budapest
June 10 - travel to Barcelona
June 11 - 13 - Barcelona

Not enough time any place - I agree to leave out Barcelona this trip.

There are lots of things you can see. What are you interested in?

Prague - Jewish Quarter, Charles Bridge, Municipal House, Mucha Museum, Castle area, Loreto church.

The only one I have been to is Prague (although I hope to get to Vienna and Budapest). What are your interests? Nightlife, churches, museums, opera, concerts, shopping?

In Prague I would make my way to Old Town Square, which is a hub of activity. The Astronomical Clock is well-known, although underwhelming IMO. You can walk across the Charles Bridge, and if you are interested in castles you should check out Prague Castle. I also enjoyed Jewish Prague, including the Jewish Museum and the Old Jewish Cemetery. I also liked the House of the Black Madonna.

Here's a link that has a lot of info re Prague:
http://www.inyourpocket.com/czech-republic/prague/sightseeing

I also like the Time Out guides:
http://www.timeout.com/prague/

She's not really double-counting. I think Adrienne and walkabout got it right, although they said it in slightly different ways.

There's also an In Your Pocket guide for Vienna.
http://www.inyourpocket.com/austria/vienna

I don't think there is an In-your-Pocket guide for Budapest. No online Timeout guide either, but there is a print one. However, there are destination guides on this website for Prague, Vienna, Budapest, and Barcelona (see top of page).

Even if you don't like opera, you might like seeing the Vienna Opera HOuse and the Prague State Opera House.

I'm too old to advise on nightlife other than to say make sure you drink plenty beer.

Thank you! I agree the time is each place is short, but i intend to come back to places that interest me the most someday

we are not into night life too, a little party yes but mostly history and monuments...
I will definitely look into the links and make a crisp plan

Don't forget to check out the 'Ruin Pubs' in Budapest!
http://ruinpubs.com/

Such very different places! 4 days in Barcelona will just be the tip of the (proverbial) iceberg but it's a good start. The highlight of my Vienna trip was Schonbrunn castle. I very much suggest you not miss that, given your interests.

I hope you enjoy Barcelona; I know we did. The architecture is what bowled us over there..and we're not really architecture buffs either. You might find something of interest for your visit in my Barcelona trip report:

http://www.fodors.com/community/europe/4-days-in-barcelona-in-feb.cfm

Two other suggestions for Prague that I wish I had known before I left: 1) Bring comfortable well-cushioned shoes. I knew that Old Town has lots of cobblestone streets, but didn't realize the other Prague areas do also. Those streets can really beat up the soles of your feet if you don't have proper footwear.

2) Learn or jot down the Czech names of tourist sites because that's what you will find on most maps and street signs. A few examples:

Old Town: Stare Mesto
Old Town Square: Staromestske Namesti
New Town: Nove Mesto
Lesser Town: Mala Strana
Wenceslas Square: Vaclavske Namesti
Prague Castle: Prazsky Hrad
Charles Bridge: Karlov Most
Jewish Quarter: Josefov
Old Jewish Cemetery: Stary Zidovsky Hrbitov
Astronomical Clock: Orloj

Also:
chram: cathedral
divadlo: theater
kavarna: cafe
kostel: church
most: bridge
Praha: Prague
trida: avenue
ulice or ul: street

There's a tourist information center near the Charles Bridge where you can go to get free info and maps. There used to be one on Old Town Square, but I don't know if it's still there:
http://www.prague-information.eu/tourist_information_center

There's a free 2-1/2 walking tour of Prague that leaves at 11:00 and 2:00 every day. It's pretty large,and may not be your thing, but just in case:
http://www.discover-prague.com/en/royal-walk-free-tour