Normally, due to their leadership ministry, in most churches the baptism of believers is done by the elders, but there is really nothing in the New Testament that forbids others from being involved in this beautiful picture of our new life in Christ and commitment to the Lord which involves not only trusting Him for salvation, but also for our new life lived on the basis of our new position in Christ as outlined in Romans 6. I think several things support this:
(1) All believers are priests of God. There is no priestly cast or distinction in the body of Christ that limits ministry to a certain few. This means not only the privilege of ministry, but the responsibility for a believer a minister (cf. 1 Pet. 2:5, 9). Some aspects of ministry are determined by gifts and role in the church like teaching and oversight of the flock, but if one has been taught the truth regarding the significance of water baptism, then he should be able to do so. I have even invited fathers to baptize their children, but not without instruction. Here is where the role and responsibility of leadership comes in.
(2) If you recall, in Acts 8:26-39, Philip baptized the Ethiopian after leading him to the Lord. He had been chosen to help in the service of the widows who were being neglected (Acts 6:5), but at this time, this was certainly not a formal office as deacon. It may show us something of how those we call deacons are to function, but they were not officially called deacons.
(3) Also, 1 Corinthians 1:12-17, Paul committed the baptizing of most of his converts to others who are not named. They are not even specified and if it were something only pastors or elders should do, you would think he would have mentioned this.
I think the same applies to serving the Lord’s supper. However, in view of the role of oversight, instruction, exhortation, protection from heresy, and leading that God has given to the elders in the New Testament, and due to the spiritual truth portrayed in baptism, it certainly should carried out only after being instructed by them and under their oversight, etc.