  The recently discovered ATi$_3$Bi$_5$ (A=Cs, Rb) exhibit intriguing quantum
phenomena including superconductivity, electronic nematicity, and abundant
topological states, which provide promising platforms for studying kagome
superconductivity, band topology, and charge orders. In this work, we
comprehensively study various properties of ATi$_3$Bi$_5$ including
superconductivity under pressure and doping, band topology under pressure,
thermal conductivity, heat capacity, electrical resistance, and spin Hall
conductivity (SHC) using first-principles calculations. Calculated
superconducting transition temperature ($\mathrm{ T_{c}}$) of CsTi$_3$Bi$_5$
and RbTi$_3$Bi$_5$ at ambient pressure are about 1.85 and 1.92K. When subject
to pressure, $\mathrm{ T_{c}}$ of CsTi$_3$Bi$_5$ exhibits a special valley and
dome shape, which arises from quasi-two-dimensional to three-dimensional
isotropic compression within the context of an overall decreasing trend.
Furthermore, $\mathrm{ T_{c}}$ of RbTi$_3$Bi$_5$ can be effectively enhanced up
to 3.09K by tuning the kagome van Hove singularities (VHSs) and flat band
through doping. Pressure can also induce abundant topological surface states at
the Fermi energy ($\mathrm{E}_{\mathrm{F}}$) and tune VHSs across
$\mathrm{E}_{\mathrm{F}}$. Additionally, our transport calculations are in
excellent agreement with recent experiments, confirming the absence of charge
density wave. Notably, SHC of CsTi$_3$Bi$_5$ can reach as large as 226$
\hbar\cdot (e\cdot \Omega \cdot cm) ^{-1} $ at $\mathrm{E}_{\mathrm{F}}$. Our
work provides a timely and detailed analysis of the rich physical properties
for ATi$_3$Bi$_5$, offering valuable insights for further explorations and
understandings on these intriguing superconducting materials.
