Advocacy & Safety - Crossing major intersections

I've just begun cycling again after sixteen years. I've been practicing in my neighborhood, and I do fairly well. I'd like to go farther, but that sometimes requires crossing major intersections. What's the best thing to do? Should I push the button on the crossing meter as if I'm a pedestrian?
I know there are some streets that have less traffic, but still it's crossing roads with four to five lanes that makes me a little hesitant.
I'd appreciate suggestions.

I guess it comes down to confidence, and it may take a while for you to build that. Building strength will also help with the speed neccesary to negotiate traffic. Judging the intentions and speed of the traffic is also a learned skill.

But your job is to signal your intentions, if turning -- especially left -- and make your move accordingly. If you are approaching an intersection and intend to turn left across multiple lanes to get in the left turn lane, you need to be doing that with plenty of distance to spare. Once you are over, hug the right side (which will often have a barrier), then take the lane in the line-up, like any other vehicle.

For going straight through ... there's nothing to it. Just proceed. If you can easily reach the walk signal, sure, give it a push.

A rear view mirror helps, but also shoulder check; it's more reliable to judge distance.

You've got to do the risk assessment. For me, it's pretty simple. When possible, I'll use the MUP. When impossible, I'll use the road. If I don't feel safe on the road, I'll use the sidewalk. If I don't feel safe on the sidewalk (or one isn't available), I'll use the car. Whenever I'm on the bike and not on the MUP or campus, I use lights, front and back. One of my main rules is to never use the road unless I'm willing to use at least half of the lane. Basically, you have to decide what you're willing to do and not do. You've only got one life. Don't die for someone else's principles.

I've just begun cycling again after sixteen years. I've been practicing in my neighborhood, and I do fairly well. I'd like to go farther, but that sometimes requires crossing major intersections. What's the best thing to do? Should I push the button on the crossing meter as if I'm a pedestrian?
I know there are some streets that have less traffic, but still it's crossing roads with four to five lanes that makes me a little hesitant.
I'd appreciate suggestions.

What is the road like that you are on? You only mention "crossing roads with four to five lanes" not actually riding on them. The easiest way to cross any major road is like you would driving your car (assuming you drive). Pull up to the light in the area where the sensor is to trigger the light. Wait until you trigger the light then proceed through on the green. This is assuming a lighted intersection of course. I know of a few intersections with 4-5 lane roads that do not have signals and you need a slightly different strategy to cross them, though again it's no different than how you would cross while driving a car.

But back to triggering those signals, there are several websites out there that discuss strategies to help cyclists trigger lights. In my area, almost all lights trigger for me simply by stopping near the stop line in the middle of the lane. There are a few stubborn signals that require me to rock my bike back and forth and side to side a bit to trigger the signal. It is rare that I come across a signal that will not trigger at all. In those rare cases, I have pushed the pedestrian signal button or waited for other traffic to arrive(usually happens pretty quickly around here).

If you arrive at the light with other traffic, my advice to you is to line up with that traffic, in the appropriate lane for your destination of course, and proceed through the light with everyone else.