Do I have to upgrade to personal computer or laptop computer in order to be able to use the IBM Quantum Experience website or run code locally?

You can run IBM Quantum Experience from any modern browser in the Tablet. In IBM Quantum Experience, you can explore basic circuits. However, you may find it difficult to use Qiskit notebooks from Tablet. You can also use Quantum Computing Playground to see how basic gates, circuits and algorithms work. Please find a list of useful Quantum Computing Simulators in this article. However, I would recommend you to use a PC or a Laptop with a minimum of 8 GB RAM to learn how to use IBM Qiskit, Google Cirq, Pyquil etc as it would help you to understand the functionalities of the Quantum Gates and Circuits better.

IBM Quantum Experience is web-based (or rather cloud-based) service, so all classical computation (i.e. circuit transpiling, simulations, showing state vectors etc.) are done on IBM servers. Your web browser serves as a viewer only. Hence you can use table or smartphone. Convenience of using is another question but it is up to you.

To run quantum computer simulation locally (for example in Python, MatLab etc.) you would need classical desktop (or laptop) because with incressing number of qubits, processing time and memory space generally increase exponentially (with exception to circuits composed only of Clifford gate - i.e. $H$, $S$ and $CNOT$ which can be simulated efficiently on classical computers).