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Purchase:
Property prices are highly regional in Germany: While a terrace house might cost EUR30000 or less in the Eastern German countryside, it is more than ten times as much in the economically successful Western cities like Munich, Hamburg, Frankfurt or Stuttgart.
Fees associated with property purchase (property tax, public notary, land registry fee) are about 10% of the sales price. If you involve an agent, he/she'll get another 3-5%.
Banks are hesitant to lend to recent arrivals with little credit history in the country. But other than that, there is no restriction on property ownership by foreigners.
Property prices are fairly stable (rising by at most a few percent per year in boom times and rarely ever decreasing during a crunch) and short-term investments in real estate are made unattractive by government rules, to discourage speculation.

Rent:
Rents also depend on the location: A spacious four-room flat of good standard can cost as little as EUR300/month in the less desirable places, and above EUR1500/month in the Western cities (for unfurnished flats - furnished places are rare and cost 50-100% more).
Most communes publish rental surveys ("Mietspiegel") at regular intervals, from which you can see the local price levels.
Rental contracts usually require a security deposit of three months rent. On top of the monthly rent, the tenant has to pay all running costs (taxes, fees, gas, electricity, insurance, etc.), which will amount to another EUR200-300 for the abovementioned 4-room-flat.
Rental contracts have a minimum termination period (by law) of three months, and the rights of tenants are very strong.

Finding places to buy or rent is easy in the less desirable places and very hard in the Western cities. Unfortunately you might not have a choice: Jobs, English speakers and high living standards are mostly in the big Western cities - that's precisely the reason they are popular and expensive!
You can only really start your accommodation search after arrival, and it might take several weeks or months - so you need a temporary place for the beginning. I recommend a hotel, hostel or serviced apartment with a weekly arrangement.

My experience has been exclusively in Berlin, so it's not at all representative for the whole of Germany of course. Anyway, when I first arrived I found myself paying no more than 300EUR for a reasonably big WG (flatshare) room in Wedding. Many others I knew were paying well over 400 for a much smaller room in Kreuzberg/Friedrichshain/Neukölln. My observation is that there's still a pretty drastic gulf in price between the highly cool and desirable areas and the not-so-hot areas. Frankly though, Wedding really isn't as far away as people sometimes like to make out, and there's an increasing amount of cool stuff to do. Personally I'm more than happy to save 100 euros a month!

When I decided to move to Stuttgart 16 years ago I could not find a decent, affordable place to rent. Overpriced or badly designed places with no direct sunlight were more common. Thus competition for nice places was overwhelming. I was already in Germany and got along with the language. But as a self-employed foreigner nobody would rent to me. It wasn't bigotry per se but when 50 people say they are interested in a place then the renter is going to go with the one they think are most reliable; so the nice German doctor/lawyer couple would always be taken over me. Thus I decided to buy a place.

I actually had money saved so that I could buy an apartment and took out a small mortgage but had enough investments to cover it as well giving security to my bank. Now my apartment is worth around 80 or 90% more than the purchase price as property has gone up in Stuttgart, especially over the last 5 to 8 years. As an investment this beat nearly any stock funds or fixed interest investments opportunity. On the other hand it is now pretty expensive for people looking to buy their first property in the area.

There are 3 options where one can buy property for a smaller initial investment but all have some major drawbacks or limitations. Thus such options are most likely beyond what a new arrival to Germany can handle but for longer term residents a possibility. One is to buy in a smallish community that is some distance to a big city. One can often find large places or a good lot to build on for little money. The reason is that many people simply don’t want to live there anymore. Especially young people want the excitement and job opportunities of a bigger city. Some places might not even have a school anymore since there are few families with kids left. And it can be very far and inconvenient to commute to a job. But for people without kids and who might work at home; like some kind of online service or job, then it can be quite attractive, especially if it’s in a pretty, quite area.

Another possibility is to buy a run-down property and fix it up. There is even a show in Germany that showcases people’s attempts to do this. If one is a professional handworker like carpenter then they might consider this. But this is a huge risk if you don’t buy the right place. One should have the aptitude to do a lot of the renovation oneself, and hopefully have contact to other handworker friends or colleges that can help or do certain tasks for cheap. One  also needs to know what they want, local market prices and to have an objective inspection of what condition the house is really in – best done by a professional. Many people underestimate the amount of work and cost and end up with a half done place and no money. Typically what looked like a few shingles needing replacement on a roof turns out to be hidden but major water damage to walls and frameworks, often resulting in harmful mold.

And even without bad surprises such a project can be overwhelming. One can do it cheaply because they are basically hiring themselves to do the work. Yet one needs to earn money in the meantime and renovate part time or somehow renovate full time and get along with little or no income. If one can renovate while the partner earns the money then it can work.  Another question is where the people live while renovating. Paying rent in addition is usually out of the question so they either have to have at least part of the property in good enough condition to live in or as I’ve known people to do, to buy a cheap, used trailer home until they can move in.

Another possibility, but again one where one needs to know exactly what they are looking for and how to go about it, is to buy distressed properties put up for auction. This is nowhere as common in Germany as in America a few years back because there was never such a property bubble here. But there are occasional property foreclosures that can mean getting a place for significantly less than the normal market value.  Beyond waiting for and pouncing on just the right place, one needs to have the financing already set up for immediate purchase. There are people who make a living buying and reselling distressed properties. They will have capital to finance things, or if needed, even take a relatively expensive credit as opposed to a lower rate, long-term mortgage. This is not a problem for them since they rely on reselling the property within some months. For one buying for a long term residency such financing rarely makes sense.

Another thing to know is that banks are relatively cautious in Germany. Mortgages are usually for 10 years, maybe 15 but not 20 or 30 years like in America. And they don’t take the risk of financing no money down purchases. Banks are usually going to want to see people pay 20 to 30 % down and have a reliable, good paying job and a decent credit rating. Probably one of the most positive factors in buying property at the moment is that interest rates are very low. And property is usually a good investment since as Beppi mentioned the markets are fairly stable and tend to go up and not suddenly drop.

Since land cost so much in medium to large German cities, few people have stand-alone homes with a big garden there. More people buy apartments which is comparable to a condominium type of ownership in America. Also popular are row houses in the suburbs where one ends up with a small garden. One has the costs and advantages of their own home but can also have issues with noisy neighbors on either side – or ones that are overly sensitive to your noise.

For people who actually have a good amount to invest, then a multiple family house is another possibility. This might consist of a larger 2 or 3 bedrooms, 100 square meter dwelling and an additional 1 bedroom 50 square meter one. A couple might buy and live in the bigger place and rent out the smaller one. If they have kids then they could eventually use the smaller one for themselves as well. Or maybe one of their parents gets old and is alone so they move them into the small apartment rather than having to put them in a nursing home.  Or one might need the second apartment from the beginning for a teenage or young adult child or an elderly parent who eventually moves out (or passes on) and then the extra space could then be rented. One would need a larger initial investment to have the additional space but rent payments can more than offset the difference at times and/or it gives flexibility for changing living circumstances.

A rule of thumb in America is that one should not buy a home that cost much more than 1 years total earnings. A nice stand along house with garden in Germany generally cost 3 to 5 times what owners earn. This is one reason that it much rarer in Germany but is also reflects that Germans tend to put more of their money into their homes. But even the ones who do this generally have some other source of money, an inheritance or parents who give or loan them the money.  Thus one needn’t question how in the world some Germans can have that nice house on what they earn – they can’t.