Can someone review my first Arduino schematic please?
Do not work with the idea that the GREEN LED and the BLUE LED can be operated without series resistor. As a starter the paralleled LEDs with lower forward voltage will starve those with higher forward voltage. Series resistors for each can even this out and take care of that problem. 

Next it is typical problem that BLUE LEDs do not always work well from a supply of just 3.3V. Consider getting your design modified to add a 5V source to supply this LED and still apply series resistors for these LEDs for the reason given above. With the problem as stated you may be able to find some BLUE LEDs that will give you satisfactory brightness from 3.3V but they will be dim in comparison to the others. 

Do not second guess the need for the bypass capacitors. Put them in by the power pin pairs of each IC chip. 

Note you failed to hook up one of the VCC pins of your MCU. It is essential to connect all power and ground pins of IC chips. 

If you want to have a charger for the battery then you should plan to put in a regular power supply circuit for your design that can accept input from the battery or from some AC-DC converter module. When the AC-DC supply is present then have the battery go into charge mode.