The Medrash Tanchuma says that Moshe Rabbeinu saw that bikurim would cease to be brought because of the chorbon Beis Hamikdash, and therefore he instituted davening Shacharis, Mincha and Maariv.
The obvious question is: What is the connection between bikurim and davening?!
There are those who answer as the Mishneh Berurah states: Tefillah must be couched with shevach and hodaa (praises and thanks) to HKBH for that which he has given to us. Also, we ask HKBH to help us in the future. In truth, these are the main elements of our Shemonei Esrei: We have shevach in the first three brochos, and hodaa in the last three brochos, and in the middle we ask for our needs. So too bikurim has within it the same characteristics (See the Devarim 26:5-10 for the words that are said when the bikurim are delivered to the Kohen in Yerushalayim).
The Jewish calendar is made of different times that catapult us into different spheres of Avodas Hashem. For example: During Pesach we work on our emunah; During Sukkos we are thankful for the goodness that Hashem has bestowed upon us; During the month of Elul, it is in our minds to take stock of that which we have accomplished and to make a commitment for the new year that it be better than the last one, etc. This cycle of mini-holiday segments repeats itself each year. Our day is also broken up into three different segments, and we interact differently during each one. There is the morning, when we are invigorated; the afternoon when we take stock of that which we have accomplished and forge further forward; and then there is the evening, when we put it all together in our minds, and put things into their proper place to use them at a later time. Even our meals that we eat reflect our attitudes to these times (In Israel, the main meal is eaten at lunchtime). Just as we make a brocho before we partake in our food, so too before we use a new segment of time we daven the appropriate Tefillah. This idea is learned from bikurim: We have planted and harvested and we are about to embark in a segment of enjoying that which we have gotten. Those first fruits that we cherish most, we give symbolically to Hashem by giving them to the Kohen. So too, each segment of the day deserves recognition that in order to succeed we need Hashem’s help.
As Rosh Hashanah nears, and we are going to make pledges to ourselves to become better people, it is proper to ask Hashem’s help that we be successful. Those first tefillos of the year are dedicated to Hashem. We don’t even ask for things for ourselves, but solely to extol that Hashem is our G-d.
With Hashem’s help as we start the year off, with our tefillos dedicated to Hashem, we will merit a year of success in our endeavors.