The Diplomatic Situation:
Burgundy starts off as one of the most prominent yet precarious nations in Europe. Holding 4 subjects, three of which are Personal unions and one of which is a Vassal; Burgundy has very limited options for diplomacy if the player wishes to not have their dip power points suffer. Starting relations with other nations can vary but the two most consistent nations with a 'friendly' attitude are Austria and Castile. This does not mean that they are the best allies however. While Austria is certainly one of, if not, the most powerful nation in Central Europe, they will not join an offensive war with France without the requisite 10 favors due to the Austrians not desiring any French land. But for the sake of keeping the Emperor amicable and for the coalition protection that having a powerful ally will incur, Austria is a fair choice. A Burgundian player's first immediate goal should be the neutering of France. For this job therefore Aragon and Castile are the best options. When the game first starts before un-pausing, check to see if either of the Iberian nations have rivaled France and check to see if either of the two nations desire French lands. This will enable a Burgundian player to promise land and start a two front war with France as soon as they wish. Due to the new Burgundian missions expansion to the south in Savoyard and Swiss territory is much more desirable. Thus, the choice of rivals should naturally fall to France,Savoy or Switzerland, and Provence. Due to all four of these nations holding land key to Burgundy's success.
The War with France:
Like with most rivals, the first war with France is the most important as it will allow Burgundy to quickly snowball at the expense of France. There are two possible war goals: the first being more risky but with greater reward and the second being safe yet profitable. The acquisition of Reims and surrounding territory would link the holdings in Burgundy with the Low Countries and gain considerable trade power in the Champagne trade node. This comes at the cost of a possible coalition. The annexation of Lyonnais would allow Burgundy to tap into the lucrative Genoa trade node. But this comes at the cost of a still halved realm and a more powerful France.
In order to make this war as favorable towards Burgundy as possible one should follow the diplomatic guide listed above and hire an 'Army Reformer' advisor for the morale or 'Military Engineer' for the fort defense. The extra morale coupled with Burgundian Traditions will allow the Burgundians to have a 20% morale boost right at the start, thus having a better army over France (who will most likely still have the better general advantage however) while the fort defense advisor coupled with the 'Defensive Edict' will net an astounding 53% fort defense to Burgundian forts, virtually ensuring no French gains will be made. The most important move to make is for the 'Surrender of Maine' event to fire. This is where one may wish to restart as for the war with France to be won very early on, England needs to refuse the surrender of Maine and declare war. If however one does not wish to restart, setting the national focus to military and demanding mil power from the nobility estate will allow Burgundy to gain the tech advantage on the French fairly quickly. Keep in mind however, that the subjects and allies of the Burgundian crown will be on the same tech level as France.
The War Itself:
If England's decision was to declare war, a few months should pass in order to get the French forces occupied with the forts in Normandy and Aquitaine before declaring war. This is where Burgundy could use their vassals in two different ways; once again, one way having higher risk higher reward, and the other being a safer, albeit costly (in terms of Attrition) strategy. The first way to use the vassals would be to set them all to siege down the province of Barrois or Paris while the actual Burgundian army goes south to help link the Iberian armies with Burgundy's. This strategy would allow Burgundy to rack up considerable warscore against France while also uniting Burgundy with their allies. But the cost is that the French forces may peel off from their English sieges and steamroll the vassal forces, leaving the Burgundian player significantly weaker. The second strategy of combining the vassal and Burgundian forces together will allow for a massive army that will have a much harder time being tackled by the French. This comes at the cost of massive attrition as this army will easily number in the 30-40k range, which is outside of the supply limit for many provinces in this time period so Manpower may become an issue. Another cost would be tactical inflexibility: Burgundy will have to pick one area and stick to it in order to make any gains there. Sieging Paris or Barrois is recommended as this will both accrue warscore (Paris especially) and allow movement from Burgundy to the Low Countries. Once these two primary objectives are completed the war with France should be taken slowly and thoughtfully, as the French still have the capacity to utterly annihilate Burgundy's forces.
The Remaining 15th Century:
Should the war with France be successful, depending on whether or not the Burgundian player has chosen to annex Champagne or Lyonnais Burgundy will either spend a few or many years to wait for AE to decay while consolidating and managing the realm in preparation for the next war with France (which should be much easier if the Champagne route was chosen). Expansion into Imperial lands is highly unrecommended at this time as the AE penalty will certainly cause a coalition war to fire when combined with the previous French conquests. Waiting for the 'Shadow Kingdom' in the 1480's to 90's is a necessity for expansion into Savoy as this will avoid the wrath of the Emperor and the AE caused by annexing HRE provinces. Also what will most likely occur is the loss of the alliance with Castile or Aragon (unless the promise for land was fufilled, but this is impossible while simultaneously acquiring the Champagne region). The vassal Nevers should be annexed by now, freeing up a diplomatic slot for more alliances. A strong Northern Italian state such as Milan or Genoa is recommended to help with expansion into Savoy and Provence. Another good choice would be an Imperial Elector such as Trier or Cologne. For if Burgundy wishes to expand into the HRE at all, they must either be the Emperor themselves or hope a weak Emperor gets elected to the position. Being elected to the Imperial Throne consistently will cause Austria to lose much of their luster. Allowing for their eventual defeat at the hands of the Burgundians and the acquirement of 'The Burgundian Conquest' achievement.
Idea groups are very often a matter of personal preference and in-game situations so recommendations are hard to make. But if the general rule of picking idea groups that further augment the National Ideas of the nation the player is controlling then Influence or Humanist should be the first idea group Burgundy should invest in. Influence would be the better of the two as this would solve two problems facing an early Burgundy game: Aggressive Expansion and Diplomatic Relations. While Humanist does help with AE, and when stacked with the Burgundian National Idea will allow for total or near total toleration of Heretics, it does not solve the diplomatic relations issue and many of the ideas in the group aren't needed until the Protestant Reformation. Military groups that come recommended are either Defensive or Offensive. Defensive would be a huge early game boon for Burgundian armies, as just the second idea of the group will allow the Burgundian Army to acquire a remarkable 25% land morale over other nations. The Fort Defense is very nice due to Burgundy having to rely on many forts because of their flat and open terrain. While the reduction in Fort Maintenance will save many ducats per month, as Burgundy starts with a large amount of forts. Thanks to the new policy system making them much more viable, Defensive offers fairly good policies when coupled with Humanist or Influence. Combing with influence will grant a minor Fort Defense boost and a rare War Exhaustion reduction per month. Offensive has great synergy with the Burgundian Ideas as this will allow Burgundy to acquire 10% discipline to augment their forces. Having generals with better fire and shock pips is always a plus no matter what situation one might find themselves in. Another reason to pick the Offensive Idea Group would be the excellent policies: when coupled with Humanist Burgundy will gain a further reduction in National Unrest and halve Years of Separatism. This makes the process of annexing newly conquered provinces much easier and rebel-free.