One of them was why would God ask us to eat His flesh and drink His blood? I remember a priest once who was talking about practices of other churches. He said we sometimes consider them weird or different. Then he says "You think that's weird? What about Catholics? We get together every Sunday and eat our God! That's weird."
The other is the prohibition in Acts of the Apostles against drinking blood. After the Council of Jerusalem, the Apostles decided that Gentiles that did not become Christian didn't have to follow the old law (specifically they didn't have to be circumcised), but they did maintain the prohibition against drinking blood. When discussing the Eucharist with Protestant friends they pointed out that the Bible doesn't contradict itself (agreed), and therefore they said that the Eucharist was a very important symbol, but a symbol. That way there is no contradiction between that prohibition against drinking blood, and Jesus' command to drink His blood. I remember discussing this with my mom. Her immediate response was, "Well, the Eucharist is an exception to that rule." Yeah, I'm going to need a little more than that.
Okay, so there were my problems. Why do I go to church every Sunday to eat my God? And why is consuming the blood of Jesus an exception to the prohibition in Acts?*
The answer to both questions was the same one, and it was one that I got from my Christian, anti-Catholic, Judaism professor in college. He had no idea that he was explaining some of the most amazing truth that I had ever heard about the Eucharist. He certainly had no idea he was answering questions that I had pondered for a couple of years at that point. Had the Eucharist actually been brought up, I'm sure he would have had a few things to tell me about why I was wrong. All he was trying to do was to explain some of the cultural aspects of ancient Jewish customs. That day, the discussion was about drinking blood.
In ancient times, there was a custom about blood that saw it as the life-force of the being, or the seat of life. This makes sense. If someone's "lifeblood" is spilled out, they die. One of the things that ancient people would do is to drink the blood of some of what they killed, believing that it would give them the life, or the strength, or the power of that being. Therefore, they might drink the blood of a bear or a bull or a lion, or even one of their enemies.
The Jews, however, with their view of one God, the creator of all things, had a prohibition against drinking blood. This is because they saw God, and God alone, as the author of life. The blood, the seat of life, was given to that one being by God. It was intended for that one being only, and it was wrong to take it from that being in order to appropriate it for oneself.
I'm probably making a hash of explaining all this, but that's when it finally all made sense. Jesus, the author of life, was giving His very life to us in a very specific, tangible, real way. No one was taking it from Him, but He freely chose to give it to us. In drinking His blood, we do not diminish or take away His life as happens when you drink the blood of a fellow creature. Rather, we receive true life. Beautiful, real, eternal, divine life.
*I always accepted that it was an exception and didn't seriously consider the possibility that the Eucharist is just a symbol. The reason is that I had already become convinced of the biblical reasons to believe in the True Presence, esp. with regards to John 6, and didn't think anything else made sense.