In beginning an adoption search, there are many good resources.
You may choose to read books on adoption and adoptees to better understand what the search will mean to you. While doing your searching, it is important to record any and all pieces of information you find, no matter how insignificant they seem. If possible, speak to adoptive family members to determine if they have any information to share with you.
You can choose to register with the International Soundex Reunion Registry. It is the oldest and largest reunion registry of its kind and it is free.
You may choose to use DNA testing that is readily available. Information shared about or by biological or adoptive family members with the adoptee may be accurate or inaccurate. DNA testing can provide the adoptee with connections to individuals with whom they share DNA. Some frequently used are Ancestry DNA, 23andMe, Family Tree DNA, and My Heritage. Please note that Ancestry DNA is not the same as Ancestry by DNA. Please research companies carefully before choosing. You may choose to use more than one. Providers often have sales on DNA testing so, if cost is an important factor, watch for these sales.
GedMatch is another DNA resource. It is free. GedMatch does not do DNA testing but allows the individual to upload raw DNA results from some DNA testing companies to the GedMatch database. It produces a larger database for you to use in your research.