You have great enthusiasm, and that will go a long way. Lithium battery technology is a very complex field.
Battery University provides credible and free information on this. I strongly suggest reading through the lithium classes. Some topics to learn about include: capacity vs voltage, charging, over-discharging, cells in parallel, cells in series. Without paralleling cells, the battery life will be very low, so make sure to look into cells in parallel.
The TP5046 is a nice elementary charge IC will charge your cells to ~4.2 using constant current then constant voltage (see
Julian Ilett's video), but the Li Ion cell voltage will drop as its capacity decreases. You risk damaging the cell if the voltage drops too low, so you want to cut off power before the cell reaches that threshold. This can be done using compactors, ADCs on microcontrollers, or through designated voltage monitoring ICs.
I noticed that you have a 655 Ohm feedback resistor, which I understand is to achieve 4.5Vout. The PTH08080 is a buck (step down) converter, which means your output cannot exceed your input voltage. The cell's voltage will drop from 4.2V as your electronics draw current from it. The regulator's output will drop linearly following the cell voltage; in fact, you won't even see 4.0V out of the regulator with a fully charged cell. Rule of thumb is that you should not demand an output voltage from the regulator that's higher than the cutoff voltage (~2.5V for li ion) of your battery.
If you want 5V output from a single cell, you'll need a boost converter like the
Powerboost 500C from adafruit. Another solution is to put 2 cells in series, which will double your voltage, and then you can use the PTH08080 to step down the voltage to a solid 5V. The trade off is that the supporting circuitry becomes more complex because now you need to balance charge and monitor the cells.
There exist ICs that balance charge cells, boost output voltage, have integrated over-discharge protection, short circuit protection in one package. They will require some active and passive components too. Their upfront cost may seem high, but they will save time reinventing many wheels, and parts that will be destroyed in the learning process. Take a look, and definitely ask questions!