I'm very involved in my church. it's not pentecostal but it is evangelical. Im not certain that it's exactly the same in terms of the offering but here's what I know in general and specifically:
Churches are supported by your offerings. Consdier how much it cost to heat yoru home and pay the elctric bill. Consider how much it costs for a much bigger church building. The pastors need to live too and need to draw a paycheck to feed, clopth & house their families. Churches also have staff - dependhig on how big yoru churhc is and what kind of programs they offer they may have a youth pastor, a children's pastor, staff to help each of those areas. Most churches haev sunday school, youth group, senior groups and also offer counseling, help out the poor in the community, hold group classes for grief, divorce, parenting, finance,etc. All of those things cost money.
The Bible calls for the "first fruits" or the best to be given to God. It also talkes about the "tithe" meaning 10%. While many scoff at such a high number, giving 10% means you live on 90%. If all you have, and all yoru earning potential is a gift from God can you afford to give him back 10%? The Bible also promises that God gives it back in abundance. I know most people do not give 10%. but those I know who do give 10% usually find their income increase.
The envelopes are provided by the church and they will usually start proviing them to you if you give your offering in the form of a check. Then they see that you are giving regularly and they know your name & address from your check. Also using either a check or the envelopes allows the church to send you a record of your giving which you can take as a deduction for tax purposes if y ou itemize on your tax return.
As for giving money to the pastor when he visits - I have NEVER heard of such a thing. I don't know the story behind this other pastor at a church you visited - but there's never an expectation of money in that instance. Perhaps the person he visited has a high income and wanted to fill a need the pastor was telling about?
Finally, in our church we only have communiion once a month - the first Sunday of the month. On that day we also take a special offering (a second one taht day only) for the poor in our chruch family. It's used to buy warm coats for families, or gas or grocery gift cards, to send a kid to camp or on retreat with the youth group, etc.
I hope you have a good church relationship. Important thing to remember as you get closer to the people at your new church - people are human and have flaws that you'll discover. If you come across people you're not sure about - pray and ask for discernment.
Have a lovely Christmas!