Stochastic Galerkin finite element methods (SGFEMs) are a popular choice for the numerical solution of PDE problems with uncertain or random inputs that depend on countably many random variables. Standard SGFEMs compute approximations in a fixed number of random variables which are selected a priori and the rates of convergence deteriorate as the number of variables increases. The size of the associated linear systems of equations also grows rapidly with the number of input random variables and desktop computer memory is quickly exhausted. In general, it is unknown a priori which, or how many, random variables need be incorporated into the discretisation in order to estimate quantities of interest to a prescribed error tolerance. Quantities of interest which depend strongly on a high number of input random variables pose serious theoretical and computational challenges and new sophisticated algorithms are required to approximate them. We focus on the design of efficient adaptive SGFEM algorithms for elliptic PDEs with inputs with affine dependence on a countably infinite number of random variables. Starting with an initial cheap-to-compute approximation, we employ an implicit a posteriori error estimation strategy to steer the adaptive refinement of the approximation space, ensuring that only the most important random variables with respect to the energy error are incorporated. To ensure that the correct decisions are made during the adaptive selection process, the error estimate needs to be highly accurate. Additionally, a suitable balance between the computational cost of the estimate and the desired accuracy must be struck to ensure that the algorithm is efficient and quick to run. This thesis contains two novel contributions. First, we investigate the so-called CBS constant associated with two finite element spaces that appears in the bound relating the true error to the estimated error. With the aim of designing cheap error estimates with effectivity indices close to one, we compute the CBS constant associated with several non-standard pairs of finite element spaces. For certain pairs, we also prove new theoretical estimates for the associated CBS constants using only linear algebra arguments. Second, we design a novel adaptive multilevel SGFEM algorithm, where each solution mode is associated with a potentially different finite element space. When applied to the stochastic diffusion problem, we demonstrate that our multilevel algorithm performs optimally in that it realises the rate of convergence afforded to the underlying finite element method for the analogous deterministic problem (despite the diffusion coefficient being modelled as a function of an infinite number of random variables). We consider convex and non-convex spatial domains, the latter of which leads to solutions with spatial singularities. To realise the optimal rates of convergence on non-convex domains, we also employ a local mesh refinement strategy within our multilevel algorithm for each solution mode.