The rule giving which verb is proper is here.
You say that « monter » is both transitive and intransitive; it's not quite so. You are dealing here with two acceptations for a given form; there is a strong similarity between the two but that's all; thus, the form "monter" embodies not just one verb but two. The definitions can be checked in the TLFi; they are and repeated below for convenience (highlighted in yellow, user LPH bold type).
I. Emploi intrans.
A. [Le suj. désigne un être vivant]
3. S'élever
b) pour être à un niveau où l'on est plus ou mieux en vue.
Monter sur le billard, sur une table d'opération; monter sur l'échafaud, sur un piédestal, sur les planches, sur le théâtre, sur le trône; monter au pinacle, à la tribune.
As the rule referred to at the beginning shows (verb indicating motion that is intransitive and therefore an exception and subject to the "house" rule (être)) you use "être".
- Je suis monté sur la Tour Eiffel pour la première fois.
II. Emploi trans.
A. [Monter implique une notion d'élévation]
1. Parcourir en s'élevant, grimper, faire l'escalade de.
Monter les marches, les degrés d'un escalier; monter un escalier, deux étages, une côte, une pente, une rampe; monter trois cordes à la force du poignet.
- J'ai monté le Saint-Gothard à une heure du matin, par une lune sublime;
As there is no exceptional case in the exemple below, you use "avoir" (according to the rule).
J'ai monté les escaliers de la Tour Eiffel quatre à quatre… au début !
The reasoning when dealing with "conduire" and other verbs is the same.
In the case of "conduire" in your example you can see it is not reciprocal, not reflexive and not in the "house". Therefore you have to use "avoir" whether the form is that of a transitive verb or not : "j'ai conduit le cheval à l'abattoir." (transitive), "J'ai conduit sur une route mouillée." ("emploi absolu"; not stated but possible: "ma voiture", "une voiture", "un camion", etc). Here, you should have checked in a good dictionary (TLFi) and you would have found that "conduire" in your example is never intransitive but that it is used in "emploi absolu" (which makes it look as if it were intransitive); however that changes nothing, as it is not in the "house". If you are talking of "conduire" in its reflexive sense, then it is different: "Je me suis conduit plutôt mal." (according to the rule).
So as to make the rule possibly simpler and more complete I'll give it the following form.
The verb "avoir" is used with all transitive verbs and all intransitive verbs except for a small list of intransitive verbs;
That list comprises intransitive the verbs that connote the idea of basic motion, for a part of them.
A
entrer
arriver
venir
accourir
monter
descendre
passer
retourner
sortir
partir
aller
tomber
B
advenir, devenir, intervenir, obvenir, survenir
ressortir
apparaitre
décéder
échoir
naitre
mourir
rester
There are two categories to keep in mind when applying that rule.
1/ The verbs connoting motion must be distinguished from more complex intransitive verbs connoting motion, as those describe a complex type of motion and as the usual verb "avoir" is used for them.
chanceler, chuter, danser, déambuler, déraper,pivoter, provenir, slalomer, skier, tituber, tourbillonner, tournailler, tournicoter, tournoyer, sauter, tressauter, trépigner,
vaciller, valdinguer, valser
2/ A second category of verbs to watch, is that of a short list of verbs that have the same form as verbs in « A » but which are transitive and have a differnt meaning; the auxiliary for the transitive form must be "avoir". Here is their list.
passer, sortir, rentrer, monter, descendre, retourner
- Elle est passé à l'étage au-dessus. ■ Il a passé son temps à la télévision.
- Le chat est sorti. ■ Elle a sorti une bouteille du réfrigérateur.
- Ils sont rentré à lécole hier. ■ La dame a rentré son linge avant la pluie.
- Il est monté dans sa chambre à minuit. ■ Il avait monté deux étages seulement pour aller se coucher.