No, a believer doesn't necessarily speak in a new language when they receive the Holy Spirit. That happened at Pentecost but it doesn't happen any more. In the similar way, believers received salvation at one time and the Holy Spirit at Pentecost, believers today receive the Holy Spirit at the same time they receive salvation. The Holy Spirit convicts us of sin, intercedes with God for us, and leads us into truth.
You shouldn't emphasize tongues when miraculous healings aren't occurring either. The lame don't walk. The deaf don't hear. The blind don't see. The dead aren't being raised either. Those things happened in the early church. They don't happen now.
Even the spiritual gifts at Pentecost were for a specific purpose in a specific time and place. They were there so people could hear the gospel in their own language. Not only to speak another language or a new language.
(Acts 2:3-11 NIV) They saw what seemed to be tongues of fire that separated and came to rest on each of them. {4} All of them were filled with the Holy Spirit and began to speak in other tongues as the Spirit enabled them. {5} Now there were staying in Jerusalem God-fearing Jews from every nation under heaven. {6} When they heard this sound, a crowd came together in bewilderment, because each one heard them speaking in his own language. {7} Utterly amazed, they asked: "Are not all these men who are speaking Galileans? {8} Then how is it that each of us hears them in his own native language? {9} Parthians, Medes and Elamites; residents of Mesopotamia, Judea and Cappadocia, Pontus and Asia, {10} Phrygia and Pamphylia, Egypt and the parts of Libya near Cyrene; visitors from Rome {11} (both Jews and converts to Judaism); Cretans and Arabs--we hear them declaring the wonders of God in our own tongues!"