Data measuring interface
Eventhough it's not fully clear what sort of an environment you are referring to by the measurement set-up, in the most broadest sense of the term you need either a data acquisition interface or a sound card for the cheapest semi-alternative.

A data acquisition interface to a PC is what you mainly need. There are a bunch of alternatives. You must set your requirements (data rate, #channels, bit depths, ADC linearity, SNR etc.) to get the optimal match for the price. Note that the software interface to the acquisition hardware is another concern that makes big functional differences in the end.

The cheapest alternative to a professional data acquisition hardware can be a PC audio soundcard. They will record AC coupled and distorted signals. So no DC is captured unles you modify (if possible) by shorting the soundcard's ADC chip's input-DC blocking caps. There are a number of soundcard capturing software which also simulate an oscilloscope as well. Most typical PC (laptop) sound cards do provide 16 bits, 2 ch, 96 kHz sampling these days.

A logic analyser is almost useless from a DSP point fo view. Logic analysers are used for digital electronics such as logic gates, microprocessors, interface protocols etc. to analyse their bits, high and low values, ones and zeros. 

An oscilloscope is again useful mainly for analog electronics diagnosis purposes. It's an indispensible hardware tool for any hardware project, but not exclusively required for data capturing, unless you want to observe in real-time what you are capturing.