Consider a spiked random tensor obtained as a mixture of two components: noise in the form of a symmetric Gaussian $p$-tensor for $p\geq 3$ and signal in the form of a symmetric low-rank random tensor. The latter is defined as a linear combination of $k$ independent symmetric rank-one random tensors, referred to as spikes, with weights referred to as signal-to-noise ratios (SNRs). The entries of the vectors that determine the spikes are i.i.d. sampled from general probability distributions supported on bounded subsets of $\mathbb{R}$. This work focuses on the problem of detecting the presence of these spikes, and establishes the phase transition of this detection problem for any fixed $k \geq 1$. In particular, it shows that for a set of relatively low SNRs it is impossible to distinguish between the spiked and non-spiked Gaussian tensors. Furthermore, in the interior of the complement of this set, where at least one of the $k$ SNRs is relatively high, these two tensors are distinguishable by the likelihood ratio test. In addition, when the total number of low-rank components, $k$, of the $p$-tensor of size $N$ grows in the order $o(N^{(p-2)/4})$ as $N$ tends to infinity, the problem exhibits an analogous phase transition. This theory for spike detection is also shown to imply that recovery of the spikes by the minimum mean square error exhibits the same phase transition. The main methods used in this work arise from the study of mean field spin glass models, where the phase transition thresholds are identified as the critical inverse temperatures distinguishing the high and low-temperature regimes of the free energies. In particular, our result formulates the first full characterization of the high temperature regime for vector-valued spin glass models with independent coordinates.