Not sure what gave you the impression Ancestry.com would be free. If you're not willing to pay any money for genealogy, you will find yourself at a lot of dead ends. Even if you don't use Ancestry.com, you will still have to pay fees for ordering individual records through the mail.
That said, start with FamilySearch.org - the biggest free records database on the internet.
Ancestry.com does also have some free records collections. This is an old-ish list but it's a start: http://ancestryinsider.blogspot.co.uk/2011/02/free-databases...
- the suggestion of seeing if your library offers access to Ancestry.com is a good one too.
Be aware that genealogy takes a lot of work - I too started out with the intent of it being a gift. I wanted to make a book of my mom's family history as a gift for her next birthday. About two years later, I finally printed the book but it quickly became outdated as I found more and more information. It's something that is always a work in progress and it may not necessarily be something you can do within a year. Some people have been working on their tree for 20+ years!
Also be wary of other people's trees who may share branches with you - they can be hugely inaccurate! So please don't think that getting info from people who have already done all the work is an easy shortcut.
You don't necessarily need to go to the courthouse in person, you can often order records online or through the mail. However, there is no one central place for this, you will need to find the individual source for each location. Some records come from state level - for example, PA death certificates come from the Pennsylvania Department of Health. You have to fill out a form, mail it off to the DOH with your fee and wait months for them to mail copies back to you. This is a big reason why genealogy can take so long - I waited 6 months for my PA death certificates to come in. Other places may have an online ordering system - sometimes they use VitalChek.
For other records, like marriage records, you may need to go to the county level. Usually, you need to look for the county clerk/register of wills/orphans court - there is typically a page for it on the county website.
There is generally a fee for getting copies of records - it can range from a few dollars to around $30 for each record depending on the source.
Good luck - if you need anymore help, just ask.