Note: there will be spoilers below!

Recap
It’s been a week since Supergirl revealed herself. She’s testing out her abilities with the D.E.O.. They want to test her if she’s going to work with them. They want to know if her powers are stable. Kara is enjoying being able to share common territory with her sister. She hears about a fire on the dock and needs to prevent a boat carrying oil from exploding. Unfortunately when she moves the ship away from the docks, the ship cracks and oil leaks out into the water. The public reaction isn’t too happy. Maxwell Lord speaks out against her.
With the current opinion on Supergirl by the public, Cat Grant's not happy. She addresses her staff in a meeting, upset that the Daily Planet has already written about Supergirl's mishap. She wants to take charge of things and interview Supergirl herself. She even asks James to arrange one with his friend in blue. When he says it doesn’t work this way, she insists someone make it happen by the end of the day. Kara talks to James about not wanting to do an interview with Cat. He does offer her some words of encouragement since she's feeling a little discouraged.

At a chemical plant, a couple of guards come across something deadly. When the body is found, the D.E.O. is called and and Alex called her sister. They figure it was another alien attack and find a strange stinger in the body.
Flashback to Krypton. Kara waits for her mom to come home. She was late because they were dealing with an alien called the Hellgrammite.
Kara tells the D.E.O. what they’re dealing with because her mother sent a Hellgrammite to Fort Rozz. Henshaw is still being hard on Kara. Alex takes Kara into a room with Kryptonite emitters. She wants Kara to be ready for anything so they practice some hand to hand combat. Kara’s sloppy and not ready. She relies too much on her strength.
Kara is talking to Cat (and actually says Superman’s name). Cat says that Supergirl is in over her head. She’s trying to take on too much at once. She should start small and work her way up. She then meets James and Winn, as Supergirl, both surprised the other knows her secret. She tells them her plan to practice and start smaller.
Some Kryptonians track down Hellgrammite. Supergirl’s aunt, Astra, talks to him. She wants to use Hellgrammite as bait so they can get her.
After Supergirl successfully takes on several smaller incidents, Alex comes over and sees that Kara told her friends about her secret. Alex is called in to the D.E.O. They figure out that the alien is actually hunting down food sources and they figure out a way to lure him in. Hellgrammite attacks, shoots his stingers, one which hit Alex. He grabs Alex and brings her back to Astra.

Cat pushes James again about using his connection to get Supergirl. She gives him 24 hours to get her in a room with Supergirl or he’s fired. Kara talks to James about doing some more superhero-ing. She also tells him she’ll do the interview. James tells her how suddenly everyone knew his name because of his friendship with you-know-who. He was hoping to escape that and be his own man in National City. She tells him how on Krypton no one is their own man. They all work together. She also says her symbol isn’t just for the House of El, it also stands for a Kryptonian phrase, their motto, “Stronger Together.” She says she doesn’t want to be a hero like her cousin, on his own. She wants to do it with her friends. When he asks if she really wants to do the interview, she says she does for what it all represents. Hankshaw calls her about her sister being taken.
Supergirl flies around, trying to listen for her sister. Alex thinks she’s talking to Alura. She says she saw her image on a Kryptonian crystal. Astra tells her that twins were rare on her planet. Kara hears her sister and knows where she’s at. She flies in rather than wait for backup. Astra attacks. Kara is surprised to see her aunt. She thought she was dead. She didn’t know her mom sent her to Fort Rozz. As Kara and Astra fight, Hellgrammite goes after Alex. She manages to make him use his stinger on himself. Henshaw arrives. When Astra grabs him, with arrogance, he stabs her in the arm with a Kryptonite knife.
Alex is recovering. Kara is dealing with finding out about her aunt. She wants Alex to help her learn to fight so they can stop her. Alex takes her to a room in the DEO base that will only open for her. It’s like her cousin’s Fortress. There’s a program made from a living memory. The program looks like her mom and is there to help her out. Kara asks about her aunt and how to stop her. Alex thanks Henshaw for helping to set up the room. When he actually refers to her as Supergirl, Alex says she won’t tell her he did that. He says thanks. As he walks away, his eyes glow red…
Astra gets the knife removed. She knows they may have to rethink and delay their time table since the humans have a weapon to use against them.
Cat is calling James about his time being up. Supergirl grabbed her car and flew it to a secluded area. Staying in the air, she says James said she wanted to talk.

Thoughts
The show is still avoiding mentioning Superman’s name. It's actually surprising when Kara says "Superman" once.

The idea of Supergirl messing up is good to see. Some have been turned off of Superman because of his all powerful nature. Supergirl may have extraordinary powers but she just started being a hero. She should need to work on using her abilities in these ways after spending so much time keeping her abilities in check. The idea of her learning hand to hand fighting skills is good as well. With a number of Kryptonians on the loose, she could be facing foes with similar abilities. She needs to have an edge over them and shouldn't rely on her powers all the time.
We do get the message of woman having to work hard to prove themselves again. As this is only the second episode, it's fine if it's brought up once in a while but let's hope it doesn't happen every week. We all know how tough and powerful Supergirl is. We don't need to be reminded every week.
It was surprising to see Kara face off against her aunt already. I thought this was going to be a slow reveal for her. It seemed she would face random Kryptonians from Fort Rozz and discover her aunt's involvement later. It's nice to see they're not holding back on this but you do have to wonder what else will be thrown at her in the coming episodes.
Hellgrammite as an alien? It's an interesting twist. It explains how and why someone with his powers is on Earth without having to spend loads of time developing his backstory. We have no idea who or what was sent to Fort Rozz. Hopefully we won't see too many other characters come out and attack Supergirl with their backstories re-written to having been sent to the Phantom Zone. It did seem odd that his stinger could be used against himself and that his body wasn't immune to the toxins within.
When Supergirl faced Astra (and other Kryptonians), it's a little surprising that they haven't developed their powers more. They should know what they're capable of doing. Even if they've been in hiding, they're likely to be aware of the things Superman is able to do.
The twist with Hank Henshaw makes things interesting. Again, it feels a little fast or sudden. We were just starting to get the impression he might be coming around to accepting Supergirl and now we have to believe he has something more sinister planned.
It's good to see the characters allowed to grow a little already in the second episode. The show is carving out its own superhero path separate from what we're seeing in The Flash, Arrow, or Gotham. There are some areas I'd like to see changed or improved but like Supergirl's character learning to be a hero, the show is learning as well.