Basically “à” can be seen as the English equivalent for “to”. At the same time “de” is used with certain verbs e.g. avoir besoin de (to need), envie de (to want)…actually more in the sense of “for” generally. When both à and de are used in a sentence, à comes before the first verb and “de” before the second one.

But try to see the different usage here:
http://www.cnrtl.fr/definition/

Also these questions might be relevant

When would one use “à” after a noun and before an infinitive?

“Jouer à” or “jouer de” for sports games?

When to use “de” before a verb in infinitive form?

I’ll edit my answer as I find some examples.

I am a french native speaker and it is “je serais heureux de continuer”. If you don’t believe me just do a Google search with “je serais heureux à continuer” and you’ll find only your question.

“à” and “de” have no equivalent in English and you’ll have to use some rules of thumb like this and that one.

For linguistic studies that try to answer this question, please see my other longer answer.
Here I summarise only the conclusion of one of the papers, which should significantly assist all French L2 learners. I do not use blockquotes to conserve readability.

_{Source: p 189, French prepositions à and de
in infinitival complements,
A pragma-semantic analysis (2008) by Lidia Fraczak, as part of Adpositions: Pragmatic, Semantic and Syntactic perspectives (2008) edited by D Kurzon, S Adler}

We hope to have shown that the “abstract” prepositions should not be excluded
from the analysis of linguistic signs as they convey differential pragma-semantic
values. The use of the preposition à in expressions containing an infinitival
complement is related to “ambivalent vision” of the fact described by the complement,
which implies simultaneous consideration of two contrary versions of the
fact, with a “transfer” from one to the other. The preposition de is compatible, in
such expressions, with “monovalent vision” of a fact. This pragma-semantic description,
apart from having a purely linguistic interest, may be of use for foreign
students learning French, by helping them to improve their skills concerning the
correlations between the preposition à or de and different lexical elements.
   Our study does not end here, as it is necessary to include not only verbal
expressions but also those whose main element is an adjective (e.g. prêt à faire, impatient de faire – “ready to do”, “impatient to do”) or a noun (e.g. capacité à
faire, capacité de faire – “capacity of doing”). Our first analysis of such expressions
allows us to consider that the opposition in terms of “ambivalent vision” / “monovalent
vision” is probably valid there as well.