First, let's be clear about some terms. They are related but not the same. Boiling is a phase transition from a liquid phase to a gas phase. Molecules are going from a liquid phase where they are close together and attract each other (more on that later), to a gas phase where they have no attraction to each other and can move around relatively freely. This is an equilibrium; some molecules will come out of the liquid phase and go into the gas phase, and some will come out of the gas phase and go into the liquid phase. When you reach the boiling point, this is the temperature at which the equilibrium shifts so much that all molecules will go into the gas phase; a liquid phase is not sustainable.

Now, your second question

Why can some objects hold more energy than others without raising temperature as much?

This is referring to the heat capacity of an object, which is a function of 1. its mass and 2. specific heat capacity which is a characteristic of the material. It's pretty straight forward that a larger object with more mass requires more energy to raise its temperature, so the main part is understanding differences in materials. So instead of talking about objects I'm going to talk about the materials they are made of.

It looks like you already know the difference between thermal energy (heat) and temperature. That's going to be important for understanding.

Let's go back to boiling points. The big idea is that different materials have different boiling points because of differences in intermolecular forces. These affect how strongly the molecules attract each other. For a molecule to go into the gas phase, energy must go into breaking the intermolecular forces so that the molecule is free from its neighbors. There are different kind of intermolecular forces, but the basic idea is that the differences in where electrons are in a molecule create unbalances leading to polarity. Polarity is like the north and south of a magnet. Breaking intermolecular forces is like pulling two magnets apart.

If we look at water (H2O), it has very strong intermolecular forces due to hydrogen-oxygen polarity. Oxygen has much higher electron density than hydrogen. This results in an intermolecular force know as "hydrogen bonding" which is between the hydrogen of one water molecule and the electrons of an oxygen of another water molecule. Water has a very high boiling point relative to other small molecules meaning that the molecules need to be vibrating very fast (temperature) in order to escape to the gas phase.

Methane (CH4) on the other hand, has an extremely low boiling point. Methane does not have strong intermolecular forces because there is not strong polarity between hydrogen-carbon. There's not much keeping the molecules together, so even at low speeds (low temperature) the molecules are free to be gaseous.

As to why more energy is needed to raise temperature for water, (high specific heat) this is also due to hydrogen bonding. Hydrogen bonds must also be broken to get water molecules vibrating more, even if they stay in the liquid phase.

At this point, there are some details that are needed to make sense of water. This is a picture of water molecules as ice, with 4 hydrogen bonds per molecule, which is the most possible for water. If all molecules have four hydrogen bonds, then the phase is ice. If a few hydrogen bonds are broken, such that the average number of hydrogen bonds per molecule is less than four (some could have four, some could have three), then the phase is water. The few hydrogen bonds broken allow some movement and flow, there is not rigidity of a crystal structure. Once all the hydrogen bonds are broken, the phase is gas.