Zurich to St Anton to Vienna questions

I have some basic Europe train travel questions. Though I have
travelled by train in the US, Japan and China, the only experience I
have with European trains is the Eurostar from London-Paris (and
back). We only had carry on bags then.

My wife and I are going to be taking a trip (early March), fly into
Zurich and then take the train to St Anton and Vienna. We will ski in
St Anton for the first 4 days, then take a train to Vienna and spend
the last 4 days in Vienna, then fly back from Vienna.

We will arrive in Zurich around 10am. We will be travelling with ski
equipment and bags. We would like to be in St Anton before the evening
for check-in and grab some dinner, that way we will be ready to ski the
next day.

I have a few questions:

1. Do I need to make train reservations before I leave for either leg
of the trip (zurich-st anton or st anton-vienna), or should I just do
it from the train station after we arrive? Was concerned if the plane
was delayed or something like that.

2. We were considering spending a few hours in Zurich, is the train
station in a convinient location for a couple of hours of sightseeing
or lunch?

3. If its possible to spend a few hours in zurich, what about our
bags? Can you check them at the station for the day, or can you check
the bags at the station early, and not see them again until we arrive
at our destination?

Thanks for any answers. We are looking forward to seeing more of
Europe and want to make the most of it. Most books we have on Europe
are just very general travel books, with hotel, restraunt and museum
info. Not as good for specifics.

Cheers,
-frankllynn