Till the first three months, your baby’s movements are fairly limited. But that should not stop you from not initiating games with your little one.
Games played during this time would be with the intention of introducing your baby to new things around him/her. Some of his/her senses like hearing would have developed by now, whereas some like visual sensation would get better with every passing day. It is good to let him/her touch and feel different objects around him/her, especially things like rattle-sound toys which give out sounds or play out music when touched.
You can also saddle your baby in your arms and dance gently with light music in the background. Apart from being a good soothing technique, the light dance movements and music will amuse your little one and make him/her feel happy.
One good past-time during this time is to read out to your baby. Although s/he would perhaps not understand what you are saying, but s/he will start picking up simple words, expression, and trick of voice modulation. Also point out to everything in the books and say out aloud what each image denotes. Once your baby develops the ability to see far-off objects too, you can play this game with more surrounding objects, and family members in the vicinity too.
Your baby’s movements develop around the fourth to sixth months. You can start introducing building blocks for him/her to play with. Give your little one the opportunity to grasp objects though s/he will still not be able to grip the small things as his/her pincer grasp is yet to develop. But you can play more physical activity-oriented games now which involve a bit of rolling over, kneeling, and crawling too. By this time, you can also make your little one get acquainted with the sense of smell by introducing him/her to different aromas like vanilla, sandal, herbal soaps, common kitchen items like coriander, mint etcetera.
This is also a good time to increase the duration of your baby’s tummy-time. Make him/her spend some time on his/her tummy, so s/he can practice lifting his head and neck region. Once s/he learns how to do that, you can also hoist your baby and hold his/her head up, but make sure s/he do so with adequate support and only when an adult is around him/her.
Around the eighth or ninth month, your baby will develop the ability to pick up small objects and transfer them from one hand to the other too. This is a good time to introduce hide-and-seek or peek-a-book like games, because by now your baby would somewhat the concept of disappearance of objects and their permanence.
By the time s/he is nearing his first birthday; your baby’s movements would become really pronounced. Your baby would pull him/her-self up and also try to lunge himself forward and walk ahead. At this time you can incentivize his/her movements by making him/her reach or cruise to his favourite place. Also mimic your baby from time to time – this will greatly entertain him/her.
Bath time can also be play time though different babies have different kinds of reaction to bathing and water. Use fancy toys, bubble baths, and aromas to make every bath experience memorable. But make sure you do not force your baby too much or s/he can end up developing a fear towards bathing in general.
These are just some general games that one can try out with their baby during the first year. As and when s/he grows, newer and more complicated games and toys can be introduced with time.