How to conceptualize the unit Temperature? For a long time, I was under the impression that Temperature is just a shorthand for the "average energy" within a system, but I discovered this is wrong (although sometimes this is a valid approximation). Temperature is not measured in joule, but it seems it is an independent fundamental unit, yet I struggle with conceptualizing what this unit tells me and what it is. I can perfectly well understand what length, weight, and time are, and although energy is famously not the clearest concept in physics either, I can understand it well enough. When I tried figuring this out I stumbled across the formula $T = \delta U/\delta S$, but since Entropy is defined in terms of Temperature, this definition does not help me further.
What does Temperature tell me about a system? How come any other basic unit can be applied to an individual particle, yet Temperature as a non-composite unit can only be applied to systems?
 A: While entropy-based definitions of temperature have their value, I would recommend against starting with them. First, you first need to grasp entropy, which is an anthropomorphic tool and difficult to grasp phenomenologically. Second, these definitions sometimes diverge from phenomenological definitions of temperature, e.g. leading to negative temperatures (on the Kelvin scale). Instead, I recommend to starting with more phenomenological concepts:
Temperature stems from the observation that if you bring physical objects (and liquids, gases, etc.) in contact with each other, heat (i.e., molecular kinetic energy) can flow between them. You can order all objects such that:

*

*If Object $A$ is ordered higher than Object $B$, heat will flow from $A$ to $B$.

*If Object $A$ is ordered the same as Object $B$, they are in thermal equilibrium: No heat flows between them.

Now, the position in such an order can be naturally quantified with a number, i.e., you can assign numbers to objects such that:

*

*If Object $A$ is ordered higher than Object $B$, i.e., heat will flow from $A$ to $B$, then the number assigned to $A$ is higher than the number assigned to $B$.

*If Object $A$ is ordered the same as Object $B$, i.e., they are in thermal equilibrium, then they will have the same number.

This number is temperature. Mathematically speaking, temperature is an order-preserving quotient space induced by the strict partial order describing the direction of heat flow between objects. Note how this is not much different from weight being constructed by ordering objects by the direction a scale tilts if you put one on each side.
Mind that all of this does not impose how we actually scale temperature: Going by the above, there are still many ways to define temperature, and any strictly monotonic function of a temperature is again a temperature. How we scale temperature comes from practical applications such as thermal expansion being linear with temperature on small scales. This is somewhat different from other quantities such as length, time, and weight, where the scale is straightforward because adding lengths, times, and weights has an apparent meaning, whereas adding temperatures doesn’t. Instead we have to look at cases where adding temperature differences has an apparent meaning, e.g., thermal expansion.

How come any other basic unit can be applied to an individual particle, yet Temperature as a non-composite unit can only be applied to systems?

Heat is kinetic energy, which can be transferred between individual particles.
However, a strict direction of heat flow only arises if we look at many particles.
And such a strict direction is the basis of defining temperature.
A: Thermodynamics and statistical physics are two equivalent but different approaches to the same types of phenomena.
Thermodynamics is a phenomenological/axiomatic approach, in terms of laws, deduced from observation of physical systems, known as laws of thermodynamics. These laws are defined in terms parameters of a system, like pressure, volume, temperature, magnetization, in terms of state functions, like internal energy and entropy, which are uniquely dependent on these parameters, and in terms of a few other quantities, like heat and work. Thus, the entropy is simply postulated as a function that is always increasing in an isolated systemw, defining the direction of evolution of physical systems, whereas the temperature is defined from the condition of equilibrium of physical systems, as
$$
\frac{1}{T}=\frac{\partial S}{\partial E}.
$$
Statistical physics is a microscopic approach to systems consisting of many particles, deriving the properties of these systems from more elementary models - like the basic classical mechanics for gases, dipole-dipole interaction for magnetic systems, etc. The phenomenological laws of thermodynamics can then be confirmed by derivation from the more basic laws, under some rather general assumptions, like ergodicity and thermodynamic limit. All the quantities of the thermodynamics are then defined differently, in terms of more basic quantities: internal energy can be represented in terms of the energy of all the particles constituting the system (kinetic plus potential), pressure is defined in terms of the average momentum transferred to the walls of the container, entropy becomes the logarithm of the number of microstates, and temperature becomes an average energy per particle (note that this is not only the kinetic energy.)
Materials

*

*This answer discusses different definitions of entropy.

*Purely phenomenological introduction to thermodynamics can be found in Thermodynamics and an Introduction to Thermostatistics by Callen (a rigorous approach) or Modern Thermodynamics: From Heat Engines to Dissipative Structures by Kondepidi and Prigogine (more introductory with lots of interesting trivia.)

*Statistical Mechanics by Huang or Fundamentals of Statistical and Thermal Physics are among the texts that present the two approaches in parallel.

*Statistical physics by Landau and Lifshitz is a well-known and reliable text, which however freely switches between the two approaches.

A: Even though we commonly measure absolute temperature in kelvin, the kelvin is not a fundamental unit as much as a made-up unit. The conversion factor between kelvin and joule is Boltzmann's constant, $k_B = 1.38\times 10^{-23}$ Joule/K, which simply says $$1~\text{K} = 1.38\times 10^{-23}~\text{J}$$
In a rational world temperature would have dimensions of energy and entropy would be dimensionless.
A: I think that it's probably best to think of temperature as being distinct from energy because of a pseudo-unit. You noted that $T = \frac{\partial U}{\partial S}$. But what are the units for $S$? Well, if you read off the Boltzmann formula for entropy $S = k_B\ln W$ then $S$ must have whatever units $k_B$ has. What are the units for $k_B$? Well, $k_B 1.38... \times 10^{-23} \,\mathrm{J}\,\mathrm{K}^{-1}$. The problem with this, though, is that it's basically the same as the old practice of defining the Coulomb in terms of Amps as $1\,\mathrm{C} \equiv 1 \mathrm{A}\,\mathrm{s}$: it's phenomenologically convenient, and probably even the most numerically accurate, but it's conceptually shaky.
Because $k_B$ isn't a measured quantity the way $S$ and $W$ are, in principle, we can actually just throw its units out (the same way we can make velocity unitless). Then the entropy starts to look more like the information theory entropy (which stole it's inspiration from the Gibbs formula for thermodynamic entropy) which has a pseudo-unit (exactly like radians). In information theory they like to use bits $S = -\sum_i P_i \log_2 P_i$, digits $S = -\sum_i P_i \log_{10} P_i$, or similar because there the concern is counting how many symbols need to be in a message to encode it in the most efficient way (the goal then being to add redundancy in a way that is uniformly robust to random errors from noise). In astronomy we'd describe entropy that uses a base-10 logarithm as a "decade" or "dex".
Being physicists, though, we like to take derivatives, and the what makes the natural logarithm natural is the way it makes taking derivatives simple (same for why we like to measure angles in radians; in isolation cycles are actually a far more natural unit for thinking about angles, and that's why we normally describe angles as $x\pi\,\text{radians}$, because $x$ is then in semi-cycles). So, what unit would entropy have then? If were were talking about the logarithm of the ratios of powers or squared field strength in waves, then the unit of choice would be the Neper, but that's not what we're doing here. Here we have ratios of numbers (probabilities), so we might as well go for $e$-folds as the unit for entropy (i.e. an $e$-fold is a factor of $e$).
Then temperature would have the unit Joules per $e$-fold, or $\mathrm{J}\,e\text{-fold}^{-1}$, just like period is technically $\mathrm{s}\,\mathrm{cycle}^{-1}$ and not just seconds.
The problem with doing this, though, is that it is extremely numerically inconvenient. A Kelvin is a useful unit of temperature, and a Joule is a useful unit of energy, that's why we use that $1.38\times 10^{-23}$ factor. Not using it would be akin to talking about number of atoms instead of moles in chemistry (another pseudo-unit, like "millions"). If we convert that to a base for the logarithm, you can get an idea for how stupendously large the combinatorics in thermodynamics are:
\begin{align}
  S &= -\sum_i k_B P_i \ln P_i \\
    &= -\sum_i \frac{R}{N_A} P_i \ln P_i \\
    &= -\sum_i P_i \frac{\ln P_i}{N_A / R} \\
    &= -\sum_i P_i \frac{\ln P_i}{\ln\left(\exp\left(\frac{N_A}{R}\right)\right)} \\
    &= -\sum_i P_i \frac{\ln P_i}{\ln\left(10^{\log_{10}\left(\exp\left(\frac{N_A}{R}\right)\right)}\right)} \\
    &= -\sum_i P_i \frac{\ln P_i}{\ln\left(10^{\frac{N_A}{R}\log_{10}e}\right)} \\
    &\approx -\sum_i P_i \frac{\ln P_i}{\ln(10^{3.15\times10^{22}})}.
\end{align}
In other words, to get temperature to have a conceivable scale we need to use a logarithm with base with a little more than $3\times 10^{22}$ digits, and it's all because probabilities tend to multiply when combined and we deal with Avogadro's number of particles at a time.
