For details on "stressed pronouns", see:
This one is case III: "When a sentence has more than one subject or object".
There's actually a very easy way to tell which is correct to say, in any situation. Whichever you would say in a singular context is the same one to use when including another person. You would not say, "Me is walking," but, "I am walking," therefore it becomes, "You and I are walking."
"I am going to the library." "He and I are going to the library."
"It belongs to me." "It belongs to you and me."
Marcher is conjugated like regular -er verbs ("parler") but manger has its own conjugation. I love WordReference.com's conjugator page - it shows irregularities in blue font. It shows that "manger" is a model verb for that conjugation (i.e. the particularity of retaining the e). http://www.wordreference.com/conj/FRverbs.aspx?v=manger
However, you did not get the reason why "mangeons" is an exception to the "marchons" rule.
The way 1st group verbs are conjugated starts with the infinitive form = march/er
Then you add suffixes: -e, -es, -e, -ons, -ez, -ent
You have probably noticed that all suffixes start with an -e, which is a "soft" vowel (as well as i), except 1st person plural (o is a hard vowel).
When consonant G is in front of a soft vowel, G gets a soft sound (like the S in "measure").
When consonant G is in front of a hard vowel, G has a hard sound (like "goat").
To keep the G its soft sound when it is followed by a hard vowel (a, o, u), an -e is added. Hence "mangeons" and not "mangons", and "mangeant" (eating) and not "mangant".
IMHO, it's just a matter of how much time you want to dedicate to learning languages. If you spend equal time studying two languages and keeping your own native tongue, you'll end up knowing them all. Practice makes perfect. It's just a matter of neurons sending electrons up pathways and each time it happens that connection is reinforced.