Did the thief go to heaven with Jesus that very day? No. In the first place, Christ Himself did not go to heaven that day. Instead he was placed in the tomb—in "hell" (Acts 2:31)—where He remained dead for the next three whole days and nights.
Second, the word "paradise" does not mean heaven at all. The word actually describes the earth when God will dwell on it. "Paradise" simply means a "garden, pleasureground; grove, park"—"a grand enclosure or preserve . . . shady and wellwatered . . . enclosed by walls." That is the definition one will find in Thayer's Greek-English Lexicon of the New Testament.
And third, in many versions of the Bible, this verse is mispunctuated. There were no commas in the Greek when the New Testament was written. So when the translators supplied the commas in the English, they were placed where the translators thought they should be. Actually, Christ did not say, "Today shalt thou be with Me in paradise," but rather, "I say unto thee today, shalt thou [or thou shalt] be with Me in paradise."
By using the word "today," Jesus was stressing the time of the making of His promise—not the time He would be in paradise. Jesus discerned the man's repentant attitude and was able to tell him that he would eventually live again, with Christ, in a beautiful world. That "paradise," however, has not yet come to this earth, and the repentant malefactor is still dead (John 3:13)!