Tucson, Arizona (CNN) -- Rep. Gabrielle Giffords was in Tucson Monday to finish what she started -- a "Congress on Your Corner" meeting that was interrupted by gunfire, a shooting rampage that left six people dead and Giffords with a gunshot wound to her head just over a year ago.
The meeting came one day after Giffords, 41, announced that she would resign from Congress this week.
"We were invited to finish what we started a year ago. I know that every person amongst us has been waiting for that moment. I told her what I wanted to tell her a year ago, and then I told her how sorry I was that I couldn't really introduce her to Christina," said Suzi Hileman, one of the roughly two dozen people invited to attend the private event.
Hileman was referring to Christina Green -- her neighbor and the youngest victim of that fateful day. Hileman herself was shot three times in the rampage.
"When I said that Christina would have grown up to be her, she took my hand," Hileman said about Giffords. "She's my hero ... She'll be back and I'll be voting for her."
Two pictures of Giffords' arrival were sent out to the public, one showing her being greeted by supporters at the event.
Another, more emotional picture was later tweeted out: It showed Giffords hugging Daniel Hernandez, the staffer credited with saving her life.
After the "Congress on Your Corner" meeting, the congresswoman toured an area food bank, where she was met with enthusiastic applause.
"A lot has happened over the past year. We cannot change that," Giffords says in a video on her website, speaking in a halting manner. She said she will resign to continue her recuperation from the brain injury she suffered when she was shot.
"I don't remember much from that horrible day, but I will never forget the trust you placed in me to be your voice," Giffords continues. "Thank you for your prayers and for giving me time to recover. I have more work to do on my recovery. So to do what is best for Arizona, I will step down this week."
Giffords, a Democrat, will submit a letter of resignation during the week to House Speaker John Boehner and Arizona Gov. Jan Brewer.
A senior House GOP leadership aide told CNN that on Wednesday the House will vote on a bill that was one of Giffords' legislative priorities -- a bill to crack down on drug smuggling at the U.S. border that is conducted by smugglers using ultra-light aircraft.
The decision to schedule a vote on the measure comes after Giffords' husband, Mark Kelly, spoke to House Majority Leader Eric Cantor over the weekend, the aide said. Kelly called Cantor to say he was "very appreciative" of his support over the last year, and mentioned that the border security bill was one of the issues Giffords wanted to address before she resigned.
The aide said leaders are now working with Giffords' office on the details, but they are planning to bring the bill up on Wednesday morning before House Democrats head off for their annual legislative retreat. A Senate version of the bill passed last year, but the House has not voted on the issue yet.
Gov. Brewer told reporters Sunday that within 72 hours of receiving Giffords' resignation, she expects to call a primary election sometime in April for political parties to pick their candidates for the state's 8th Congressional District seat.
That would be followed by a general election -- likely in June, according to the governor. The winner will head to Congress for the rest of Giffords' term, which ends in January 2013.
"Her recovery has been nothing less than a miracle," said Brewer. "Who knows what's going to happen in the next two years? I don't believe we've seen the last of Gabby Giffords."
Before she steps down, Giffords will attend the State of the Union address Tuesday in Washington, her office said.
Rep. Debbie Wasserman Schultz of Florida, the Democratic National Committee chair, told CNN that Giffords -- a close friend -- came to the conclusion that she couldn't devote her full energies to both her job and her recovery.
The congresswoman's recent return to Tucson for the one-year anniversary of the January 8 shooting showed her "how challenging it would be. It came into focus then," Wasserman Schultz said.
"This decision is really quintessential Gabby Giffords," she added. "She needed to step aside for herself, so that her constituents could get full-time representation. She cares about them so much, that was her No. 1 priority -- taking care of them."
President Barack Obama was effusive in his praise for Giffords, saying she "embodies the very best of what public service should be." He added that her "cheerful presence will be missed in Washington (and) she will remain an inspiration to all whose lives she touched -- myself included."
"Over the last year, Gabby and her husband Mark have taught us the true meaning of hope in the face of despair, determination in the face of terrible odds, and now -- even after she's come so far -- Gabby shows us what it means to be selfless as well," Obama said in a written statement.
In her website video posted Sunday, Giffords says her public service career has not ended.
"I will return, and we will work together for Arizona and this great country," she says.
Earlier this month, Giffords joined thousands for a vigil in Tucson, one of several emotional events to mark one year since the shootings. In addition to those killed, 13 people were wounded in the attack.
Giffords' right side remains weak. She has made few public appearances since the incident, with some rare exceptions, such as casting a vote in Congress to raise the federal debt ceiling and conducting an interview with ABC's Diane Sawyer.
She has been undergoing intensive rehabilitation in Houston, but returned to Tucson several times, according to her office.
Prosecutors accuse Jared Lee Loughner, 23, of carrying out the attack, which purportedly targeted Giffords.
Loughner could face the death penalty if convicted on charges of killing six people -- including the chief federal judge of Arizona, John Roll. Loughner has been diagnosed as schizophrenic and has spent time on suicide watch while in custody. He is undergoing treatment in Springfield, Missouri.
A federal appeals court in May cleared the way for him to be forcibly medicated, over the objections of Loughner and his attorney.
Giffords worked at PricewaterhouseCoopers and as CEO of El Campo Tire, her family's business, before being elected to the Arizona House of Representatives in 2000. Two years later, she became the youngest woman elected to the Arizona State Senate.
She won a November 2006 election to Congress, and has won two terms since. Kelly, her husband, is a retired Navy captain and astronaut.
CNN's Paul Vercammen, Deirdre Walsh, Tom Cohen and Dana Bash contributed to this report.