Photoelectron spectroscopy employing X-ray and extreme ultraviolet (XUV) radiation is one of the most important experimental methods to study the electronic structure of atoms, molecules, and solids. Recent developments of XUV and X-ray sources with ultrashort pulse durations, like free-electron lasers (FELs) and high-order harmonics of infrared lasers, enabled combining this approach with a concept of a time-resolved measurement, where a pair of synchronized short light pulses is used to initiate and observe a physical or chemical process of interest. Among other advances, such combination turned out to be particularly useful for atomic physics and gas-phase femtochemistry, where femtosecond or even sub-femtosecond short-wavelength radiation can be used to trigger the dynamics in high-lying states previously inaccessible for time-resolved measurements and offers a variety of novel schemes to probe light-induced electronic and nuclear motion. One of the key challenges for time-domain studies employing short-pulsed radiation sources is that they are necessarily broadband and, thus, typically populate a broad range of atomic of molecular states. The main goal of this thesis is to develop an experimental approach that enables state-selective analysis of the dynamics induced by such broadband femtosecond pulses in the XUV domain, and to apply it to study several exemplary reactions in photoionized molecules. Since reducing the bandwidth of the XUV pulse would ultimately limit the achievable temporal resolution, in this work the challenge of state selectivity is addressed by employing photoelectron-photoion and photoion-photoion coincident measurements. In the experimental apparatus developed as a part of this thesis, a double-sided velocity map imaging (VMI) spectrometer for coincident detection of electrons and ions is combined with a femtosecond pump-probe setup that includes a near-infrared (NIR) laser and a fiber-based XUV source based on high-order harmonics generation. This instrument has been commissioned, characterized, and applied to several time-resolved experiments on atomic and molecular targets. More specifically, this thesis describes three different sets of experiments. First, a brief overview of several XUV-NIR pump-probe measurements addressing two-color single, double or triple ionization of atoms is presented. Here, the main focus is set on capturing generic characteristic features of the corresponding two-color signals, and on revealing physical mechanisms determining their "transient" or "steady" behavior with respect to the time delay between the XUV and NIR pulses. The second series of experiments focuses on exploring coupled electronic and nuclear dynamics in XUV-ionized CO2 molecule probed by the synchronized NIR pulse. This study, which constitutes the central part of the thesis, relies on the detection of the photoelectron that reveals which electronic state is initially populated, in coincidence with ionic fragments, which provide information on the specific dissociation channel of the molecular ion after the interaction with both pulses. Here, we observe signatures of an electron-hole wave packet motion near a conical intersection of two low-lying cationic states, trace rotational dynamics determined by the dependence of the state-specific XUV photoionization cross section on molecular orientation, and disentangle the contributions of individual states to different dissociation pathways. The third series of experiments aims at studying nuclear dynamics in XUV-ionized alcohol molecules, focusing on the channels involving ultrafast hydrogen motion. Here, ion mass spectrometry measurements on methanol and its deuterated isotopologue CH3OH and CD3OH show that, depending on a specific XUV wavelength, the formation of molecular hydrogen or trihydrogen cations can be either dominated by the channels combining the hydrogen from the oxygen site with one or two hydrogens from the methyl carbon, or by the ejections of all hydrogen atoms from the methyl group. Coincident electron spectra for specific ionic fragments enable linking these channels to the calculated dissociation pathways leading to H2+ or H3+ formation. Finally, we present the results of XUV-NIR pump-probe experiments on ethanol, where a transient enhancement of particular dissociation channels has been observed. The experimental methodology presented in this work can be readily extended to a broad range of molecular systems, including both, molecular ions and high-lying excited states of the neutral molecules. At the same time, highly-differential data on small polyatomic molecules like CO2, methanol, and ethanol presented here, can be used to benchmark theoretical models for XUV ionization of these prototypical systems, improving our general understanding of light-induced molecular dynamics.