You can download PAF for free from the Mormons. You may not agree with their theology (I don't) but they won't trick you and it won't have a virus. PAF has tens of thousands of users. I read "no download"; I'm arguing that you should. If you do, your data will be on your PC where you have control over it. PAF is to genealogy research what "Word" is to writing a novel; it doesn't do the work for you, but it makes it 100 times easier than using a pencil and paper.
There isn't any such thing as a "free" site". They all cost someone money. Some are free to you because a church or genealogy society or individual or government organization is footing the bill. Some are free to you because they throw ads in your face and the advertisers foot the bill. Some are subscription sites.
If you find a site that stores your data on their server for free, there is nothing to say they can't wait until you have 300 hours of research invested in your work and then either sell it to others or tell you they are going to have to start charging you. If your 300 hours of research are on your PC, they can go whistle.
There are over 400,000 "free to you" sites devoted to genealogy. There are over 3,000 resolved questions of the form "How can I find my family tree, for FREE?", too. The answers are full of links and tips.
99.99% of you kids lose interest when we tell you genealogy research is work, a bit like doing a term paper in history class. Also when we say it will take hours, not minutes. If you don't lose interest, write to me via my profile with the names of a parent and child or husband and wife who were alive and living in the same house in the USA in 1930 - even better, husband, wife and two children - and I'll look for them in the 1930 census to give you a head start. I'll need name, birth year and birth state.
Most of us around here don't recommend storing your family tree information on websites, free or otherwise. It's much better to use a genealogical software program on your own computer to organize and store data for a variety of reasons. When you upload/store to a website, the information becomes their intellectual property. Also, it can create a situation for you where you either have information missing or violate privacy laws and etiquette standards by posting information about living people.
Some of the websites allow you to upload from genealogical software. This requires a download or installing a copy purchased at an electronics/office supply store. It also usually doesn't allow direct editing. All you can do is upload a new copy. Others allow you to input directly into the website (like ancestry.com). Most of these websites are for a fee.
The only free option is to download free software. Free software (like PAF) is available at https://familysearch.org/products
The only option without a download is to purchase a copy of the software from a software dealer. Stores like Best Buy, Staples, and Office Max sometimes carry packages like Family Tree Maker.
None of these help with research. They store and organize the information so you can be productive in your research. To research use one document at a time, one person at a time, one generation at a time. There are tens of thousands of free websites and offline resources. Which resource is most helpful will depend on where the document you're looking for is stored, which depends on where and when your ancestor lived, what records they left behind, and where the record can be found today. Not everything is online and/or free. Attempting to use just one free website will leave a lot of holes in your research.
Some helpful free starting places:
http://www.cyndislist.com/ (START with How To and Genealogical Standards and Guidelines)
Here -- Many of us have resources or knowledge specific to certain documents, times, places, and groups.
Google -- Look for local libraries, archives, agencies, and GenWebs, or websites similar to above for your ancestor's country.
Offline -- Libraries, archives, museums, genealogical societies, Family History Centers (operated by the Mormons), etc. BIG note: FHCs and some libraries have free access to some of the paid subscription sites like Ancestry.