What are the $|+rangle$ and $|-rangle$ states?

The $|+⟩$ and $|-⟩$ are states given by the following decomposition in the Z-basis: begin{equation} begin{aligned} |+⟩ &= frac{1}{sqrt{2}} Big(|0⟩ + |1⟩Big) |-⟩ &= frac{1}{sqrt{2}} Big(|0⟩ - |1⟩Big) end{aligned} end{equation} As from quantum mechanics, you can see that there's a 50% chance for both states to be found in the |0⟩ or |1⟩ state when measuring in the Z-basis.

The terms Z-basis and X-basis refer to the two pair of states ${|0⟩,|1⟩}$ and ${|+⟩,|-⟩}$ respectively, which are eigenstates of the two Pauli matrices $sigma_z$ and $sigma_x$ (or in IBMQ terms, the Z and X operators). They're called Z and X basis for simplicity and in the Bloch sphere are usually represented on the corresponding z and x axes, but there's no connection with the real space directions and in other quantum mechanics fields are referred to as 1 and 2, with no indication of axes. The Hadamard gate is just given by this matrix in the Z basis begin{equation} frac{1}{sqrt{2}} begin{pmatrix} 1 & 1 1 & -1 end{pmatrix} end{equation} which converts the state $|0⟩$ to $|+⟩$ and the state $|1⟩$ to $|-⟩$. This is all you need to know about what it does, no need to get in depth about Clifford algebra and gates which might be a bit overkill.

Turning to the other questions, it's kind of hard to explain. QC is not just another programming paradigm or language, but a totally different way of performing computation. It is still in its infancy, we're at the "circuit model" of computation and none of the hardware nowadays available can perform the few algorithms (Shor, Grover, Deutch-Jorsza etc.) on more than a couple qubits. For these reason and a couple thousands other, QC is still mostly a research field and you need a lot of math to understand it, since any computation effort is quite strongly dependant on the hardware and thus on the quantum mechanics behind it. At the very least, you should know a bit of linear algebra.

As an example, think of the classical computation in its earlier days of turing machines and circuit model of computation. To understand these you needed quite a bit of math (though a different kind of math from QC, mostly combinatorics and logic). Nowadays, though still taught at university, nobody needs to know anything about circuits or Turing machines in order to program, nor it has to worry much about the hardware performing the computation.

Edit: I think the best route for you to approach QC would be to take part in the qiskit development, it is hosted on github at this link.