Environmental monitoring robots often need to estimate data fields (e.g., salinity, temperature, bathymetry) under tight resource constraints. Classical boustrophedon lawnmower surveys provide geometric coverage guarantees but can waste effort by oversampling predictable regions. In contrast, informative path planning (IPP) methods leverage spatial correlations to reduce oversampling, yet typically offer no guarantees on estimation quality. This paper bridges these approaches by addressing IPP with guaranteed estimation uncertainty in complex environments: computing the shortest path whose measurements ensure that the Gaussian process (GP) posterior variance -- an intrinsic uncertainty measure that lower-bounds the mean-squared prediction error under the GP model -- is upper bounded by a user-specified threshold over the monitoring region. We propose a three-stage approach for efficient environmental monitoring: (i) learning a GP model from prior information; (ii) transforming the GP kernel into binary coverage maps that identify locations where uncertainty can be reduced below a target threshold; and (iii) planning a near-shortest route to satisfy the global uncertainty constraint. Our approach incorporates non-stationary kernels to capture spatially varying correlations in heterogeneous phenomena and accommodates non-convex environments with obstacles. We provide near-optimal approximation guarantees for both sensing-location selection and the joint selection-and-routing problem under a travel budget. Experiments on real-world topographic data demonstrate that our planners achieve uncertainty targets with fewer sensing locations and shorter travel distances than representative baselines. Furthermore, field experiments with autonomous surface and underwater vehicles validate the real-world feasibility of the approach. Our code is available at: www.sgp-tools.com

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