Hi Mommy!
Congratulations! You’ve just gotten past the newborn phase! In this second month, you’ll get to notice the reason behind your baby’s giggles and grunts. This is the stage where your baby’s personality is kind of showing off and how much of an active communicator she is! 
Compared to last month, her actions now are more definite and conveys meaning.
Here are some cues that your baby is saying “I’m interested” 

• Looking at your face
• Smooth movements of arms and legs
• Reaching out to you
• Turning eyes or head towards you
• Smiling, cooing, and having general bright, happy expressions

Or “I need a break”

• Turning head and eyes away
• Crying, fussing
• Coughing
• Back arching, squirming, and pulling away
• Blushing of skin
• Yawning, frowning. Hiccups

Aside from those cues, your baby continues to show off different motor skills. It means that she already has more control over her body. They could hold their head a little steadier while lying on their tummy which means stronger neck! Sucking would be very prevalent on the second month of life, too. You may notice that she sucks her own thumb, or your thumb, the blanket or the pillow case more often than not. You may wonder why but that’s your baby’s way to sooth and comfort herself. She also loves to grasp a toy near her and watch colorful, moving objects hanging in front of her. 
What better way to reward yourself than to witness a toothless, innocent smile from your little one! Yes, this time marks the beginning of smiles and more smiles from her! What a momentous milestone that really deserve a photo op! So grab those cameras and take a photo while those gums are showing off for free. 
Most of all, sob stories wouldn’t be her only way to communicate to you! Finally, here comes a few giggles, grunts and sweet coos and other various sounds that correspond whether she is hungry, sleepy, gassy or lonely and in need of some attention. But of course, teary tales would still be the number one way she would communicate his likes and dislikes. 
Often before the end of the second month, baby likes to be carried upright, facing out; she’d be demanding this way of being held. Baby’s vision is becoming clearer and she can now see interesting and exciting sights around.  Mommy, a carrier is your best friend (aside from wipes)! Baby loves to be inside a carrier while taking a stroll in parks or open spaces. As her senses develop, it would be a good opportunity to take her out into the world with you. She is seeking ways to adapt and respond to this bigger world and pay attention on how little and how much stimulation she could handle. 

Hope our lists ticked off some of those that are written on your development book!
Source: 

Your Baby’s 2nd Month.Baby Development:http://goo.gl/HlFKUt

Your 2 month old. http://goo.gl/HlFKUthttp://www.webmd.com/parenting/baby/baby-development-2-month-old?page=2

Baby’s Second Month: A guide to development and Milestones
https://www.parentmap.com/article/babys-second-month-a-guide-to-development-and-milestone