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Orange & SFR are the 2 main players, there are others but i have been with both these companies in the 4 years i have lived over here. Of the 2, i have found Orange to have better coverage both in rural towns and the bigger cities.
We pay 46€ a month for internet, orange tv, a landline phone and a mobile phone, the landline has unlimited phone calls to most countries and the tv box can be set up in English so films play in the original language they were filmed in. Mobiles and Internet packages vary immensely and you will need to put some legwork in and compare pricing.
Hope this helps a bit.

As of 2015:

How to proceed to get a landline installed in France?

You have options to choose from Orange, bouygues, SFR, Free and a couple others (I dont remember the names). You contact them and they will make the necessary appointments to install the cable etc. Usually, this service comes along with the internet connection. Note that some of the packages are internet phones, so you will get a router which splits the signal into phone line and internet.

The cheapest is bouygues which is 19.99€ which is only phone and internet. Other services offer TV also.

What are the mobile operators?

again the same operators. Cheapest is Free 2€ option. I am using their 19.99€ option which allows roaming in some European countries.

What is your average monthly budget?

Cheapest is 19.99(bouygues)+2(Free). For a decent option you should have a budget of 40€.

I was using free but found their coverage not so good and had issues with the internet connectivity too.. I switched over to orange.. the package starts from 36euros. Of course free is cheaper than orange starting at 19.99 euros. It does take about ten days to get your connection at free.. and orange was  more quick on that part.

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As a small user and I travel back and forth regularly I use Giffgaff on a sim only deal and top up automatically as and when required. I find I spend about one fifth of what I did before with Vodafone. The only drawback is that I do not connect data roaming as it can be expensive but that will all change next year when the new laws come in.

Often, a landline is included with your internet package. This has been the case with my internet with SFR and Free.

I've had Free Mobile for over three years and I think they are great. I can call the US for free, and now they have a travel package where you can use for phone and data in certain European countries for free for up to 30 days (for each country). I pay 19,99/month for a plan with 3G (15,99 now that it's linked to my internet account with Free) and before that I had a basic plan with no internet at 2 euros/month. I've had to deal with SFR customer service for a former employer and I hated them.

I chose to get Freebox internet because I've been so happy with Free mobile, but internet is a separate division of Free, I found out, and their customer relations team is wildly incompetent! Opening the account and getting the internet installed was laborious, and they never took responsibility for their mistakes or compensated us in any way, even when we had been previously promised compensation. We pay 30 euros a month for internet and the landline is included (free calls to other landlines).