To get a nice overview of how and why Arakelov theory started you could read the introduction to R. de Jong's Ph.D. thesis on
http://www.math.leidenuniv.nl/~rdejong/publications/
I remember that being very helpful to me.
To avoid too many complex analytic difficulties you should stick to the case of arithmetic surfaces (i.e. integral regular flat projective 2-dimensional $\mathbf{Z}$-schemes). The complex analysis involved is all "Riemann surfaces theory". An elementary and thorough treatment of this is given in P. Bruin's master's thesis
http://www.math.leidenuniv.nl/~pbruin/
Arakelov theory provides an intersection pairing on an arithmetic surface $X$. The idea is to add vertical divisors on $X$ above the "points at infinity" on Spec $\mathbf{Z}$ (or Spec $O_K$ ). There will be two contributions: finite and infinite. To get a good understanding of the finite contributions I recommend reading Chapter 8.3 and 9.1 of Q. Liu's book.
I remember that after reading these texts the article by Faltings was much more readible to me. I also enjoyed the very nice asterisk by Szpiro on the subject, Séminaire sur les pinceaux de courbes de
genre au moins deux (all in French, last page has an English abstract).
Here's some advice on what you shouldn't read when you just start. I wouldn't start immediately reading the papers by Gillet and Soulé (unless you really want too). The complex analysis is very involved. The paper by Bost "Potential theory and Lefschetz theorems for arithmetic surfaces" introduces the most general intersection theory (based upon $L^2_1$ Green functions) and should also be left for later reading in my opinion.
To learn Arakelov theory the proofs don't really help me understand the statements for they are based upon moduli space arguments usually (e.g. the proof of the Noether formula). Therefore, I would also recommend you skip most of the proofs on a first reading.
What did help is seeing how Arakelov theory gets applied. I recommend the recent book by Couveignes, Edixhoven, et al. available here
http://arxiv.org/abs/math/0605244
Best Answer
This may be more suitable as a long comment. I remember someone asking Soulé current open problems in Arakelov theory during a walk at the summer school (2017) in Grenoble. The adelic intersection theory is one of the topics he mentioned. The other topic is the questions left out in Arakelov's ICM talk. He said many of these are still open problems.
However, I am not familiar with the recent literature on this topic (to be specific, work by Yuan and Zhang, ACL, Gubler, etc). Berkovich spaces showed up a lot in recent literature so that we may work with archimedean and non-archimedean spaces in the same setup. Since the main difficulty faced in Arakelov theory is from the places at infinity, I do not know what is the essentially new contribution to the theory from the adelic point of view. But I am also quite ignorant.
My naive understanding is that because there is no "natural metric" on the whole adele ring, the classical Arakelov framework becomes much harder to work with. The points are "thickened" and the $p$-adic analysis analog of objects at infinity may not be available (Zhang has a paper Admissible pairing on a curve exploring this, before the invention of tropical geometry!). For one-dimensional Arakelov theory, there is a theory of matrix divisors by Ichiro Miyada. There is also some possibility of extending it to the functional field. But for higher dimensions I do not know any serious attempt to generalize Faltings-Riemann-Roch to the adelic setting.