What is the best way for a startup to open a new office and expand to a new country in Europe (company structure, hiring process, tax questions, etc)?

It depends on the country. My best advice to you is to find a key person,a local one, in that country whom you must be able to trust 100% to take care of everything.

Use virtual office solution like www.Regus.com , use a local recruiting service or the best option is hire from a competitor.

Taxes and structure you need to consult a local resource or many countries have programs to help companies expand overseas. You may even have a local resource at one of your city chambers.

I successfully ran business development for a Global Company in Western Europe and I currently work for a large European software company with offices in several European countries. I currently live in Silicon Valley.
From what I can tell, your company has no presence in the region. Are you looking to start a representative office, operations ...?
Since it is a startup, an acquisition may not be an alternative.
Depending on the reasons to expand in Europe, an option may be a BOT (build, operate, transfer) with a local company. As you know, starting from scratch will present several challenges (e.g. hiring, incorporation, etc.)
I will be glad to have a call with you to discuss alternatives depending on your business strategy.

First, and foremost, it depends on your business, industry and type of investment. Lets assume you want to open a large business to break into the Russian market....we'll go with Russia, as that is what I'm most familiar with.

First of all, you need to do market research. Russian tastes and attitudes are unique. In some ways they may seem similar to other European/Western nations but its its own combination. So you need to hire a company to do proper market research.

Suppose all that is done, now what? Well, now if this is your first time doing business in Russia, you need to find a qualified partner. Take a 40% minority stake with a right to buy out in x years. Russia, like any country you are not familiar with, has its own ups and downs and learning to navigate all this while trying to get your business off of the ground is a hard way to learn some possibly hard lessons.

So the best way is to have a local partner, someone who is experienced and has the right connections.