I read in the gospels about the Lord's prayer, but I also see that Christians do also do lots of other prayers and even have an ongoing dialogue or prayerful listening with God.
✝ Firstly, I am wondering about prayers for others... If you do a prayer for another person can that help? If another person asks you to pray for them is that ok and helpful, or is it wrong?
We are encouraged in Scripture to pray for one another, to pray for our neighbors, pray for those in positions of power, etc. It's important to not think that prayer is some kind of "magic" whereby if we say the right words, or invoke God's name that somehow we gain power or can use God like a genie or magic lamp. But what prayer does is it invites us to submit ourselves to God, it invites us to change the way we think, speak, and look at the world by pointing us away from ourselves and to the will of God.
When we pray for another person, we are saying, "These things are in Your hands, Lord; we trust You and Your ways".
Prayer is important, not because through prayer we bend the world to our will; but rather through prayer we ourselves are the ones being changed, we are the ones being shaped by prayer. Prayer changes us. Which is why how we pray has always been an important idea in Christianity, Jesus Himself teaches us that when we pray we should not pray boastfully or as a spectacle to make ourselves look better before others; but teaches us to pray through the Lord's Prayer. And the ancient axiom of the Christian Church has always been Lex orandi, Lex credendi--the law of prayer is the law of belief. What and how we pray (and worship) influences, affects, transforms, and shapes what and how we believe.
✝ Secondly, some Christians make appeals to major saints, to Mary etc. for these saints to intercede with God in their behalf.. Is that the same sort of thing, or is it wrong?
This depends on who you ask. Generally speaking only Orthodox and Catholic Christians petition the saints in heaven to pray for us, you might find this also among some Anglo-Catholics within the Anglican Church; but generally it's something only Catholics and Orthodox do.
Many Protestants believe it to be deeply wrong, even sinful, to ask the saints to pray for us and their argument for why varies.
For Lutherans, such as myself, the reason we don't petition the saints in heaven to pray for us is because we can't be sure that the saints can hear us. Maybe they can, maybe they can't, we just don't (and can't) know. And so even though asking the saints to pray for us has been a very old practice within the Church, Lutherans have generally avoided the practice because there's no way we can really know or not. However, this does not mean that we don't believe the saints can and do pray for us, indeed, we believe that the saints in heaven pray for us, even as the angels pray for us (indeed, even Scripture itself shows angels praying for God's people).
So I guess the answer is..it depends.
-CryptoLutheran