Most of the Old Testament seems directed to the nation of Israel. What part of the Old Testament is applicable for Gentile believers?
First, the Gentiles had little or no relationship to God in the years prior to Jesus' incarnation, as Paul explains:
Prior to the advent of the Church age at Pentecost, the Gentile world was largely excluded from the grace of God. While a few Gentiles were offered opportunity to enter into the assembly of God’s people in Israel (i.e., Ruth, Rahab, etc.), the vast majority of Gentiles were “excluded” from the commonwealth of Israel.
Once Christ appeared, the New Covenant was inaugurated in His blood, and all believers (Jew and Gentile) are united in one body by faith during the Church age.
Nevertheless, all who have been united by God’s grace in the New Covenant benefit from the entirety of God’s word, both Old and New testament. All of God’s word is beneficial to our spiritual growth and relationship with the Lord. As Paul says:
While many commands found in Old Testament scripture will not apply directly to Gentiles (i.e., the Law of Moses), nevertheless study of these scriptures still benefits a Gentile believer in the New Covenant. Among other benefits, we learn about the work of God in previous ages and with other peoples, we learn about promises made concerning future events, we learn God’s standards for holiness, and we learn about Christ, Who is pictured in these things.