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First moments of GL(3) × GL(2) and GL(2) 𝐿-functions and their applications Let 𝐹 be a self-dual Hecke-Maaß form for GL(3) underlying the symmetric square lift of a GL(2)-newform of square-free level and trivial nebentypus.In this paper, we are interested in the first moments of the central values of GL(3) × GL(2) 𝐿-functions and GL(2) 𝐿-functions.As a result, we obtain an estimate for the first moment for 𝐿(1∕2, 𝐹 ⊗ 𝑓 ) in a family, where 𝐹 is of the level 𝑞 2 , and 𝑓 ∈  * 𝑘 (𝑀) for any primes 𝑞, 𝑀 ≥ 2 such that (𝑞, 𝑀) = 1.We prove the subconvex bound for 𝐿(1∕2, 𝐹 ⊗ 𝑓 ) involving the level aspects simultaneously in the range 𝑀 13∕64+𝜀 ≤ 𝑞 ≤ 𝑀 11∕40-𝜀 and 𝑀 > 𝑞 𝛿 for any 𝜀, 𝛿 > 0 for the first time.Moreover, we further investigate the first moments of these 𝐿-functions in the weight 𝑘 aspect over 𝐾 ≤ 𝑘 ≤ 2𝐾, with 𝐾 being a large number.As the results, we obtain a Lindelöf average bound for the first moment of 𝐿(1∕2, 𝑓 )𝐿(1∕2, 𝐹 ⊗ 𝑓 ) of degree 8 and an asymptotic formula for the first moment of 𝐿(1∕2, 𝐹 ⊗ 𝑓 ) with an error term of 𝑂(𝐾 -1∕4+𝜀 ), respectively.